PROOF OF THE 1-FACTORIZATION AND HAMILTON

arXiv:1401.4164v1 [math.CO] 16 Jan 2014

PROOF OF THE 1-FACTORIZATION AND HAMILTON DECOMPOSITION CONJECTURES II: THE BIPARTITE CASE ´ ¨ BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN

Abstract. In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′ (G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ ≥ n/2. Then G contains at least regeven (n, δ)/2 ≥ (n−2)/8 edge-disjoint Hamilton cycles. Here regeven (n, δ) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. According to Dirac, (i) was first raised in the 1950s. (ii) and the special case δ = ⌈n/2⌉ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible. In the current paper, we prove the above results for the case when G is close to a complete balanced bipartite graph.

Contents 1. Introduction 1.1. The 1-factorization conjecture 1.2. The Hamilton decomposition conjecture 1.3. Packing Hamilton cycles in graphs of large minimum degree 1.4. Overall structure of the argument 1.5. Statement of the main results of this paper 2. Notation and Tools 2.1. Notation 2.2. ε-regularity 2.3. A Chernoff-Hoeffding bound 3. Overview of the proofs of Theorems 1.5 and 1.6 3.1. Proof overview for Theorem 1.6

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Date: January 17, 2014. The research leading to these results was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement no. 258345 (B. Csaba, D. K¨ uhn and A. Lo), 306349 (D. Osthus) and 259385 (A. Treglown). The research was also partially supported by the EPSRC, grant no. EP/J008087/1 (D. K¨ uhn and D. Osthus). 1

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´ ¨ BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN

3.2. Proof overview for Theorem 1.5 4. Eliminating edges between the exceptional sets 5. Finding path systems which cover all the edges within the classes 5.1. Choosing the partition and the localized slices 5.2. Decomposing the localized slices 5.3. Decomposing the global graph 5.4. Constructing the localized balanced exceptional systems 5.5. Covering Gglob by edge-disjoint Hamilton cycles 6. Special factors and balanced exceptional factors 6.1. Constructing the graphs J ∗ from the balanced exceptional systems J 6.2. Special path systems and special factors 6.3. Balanced exceptional path systems and balanced exceptional factors 6.4. Finding balanced exceptional factors in a scheme 7. The robust decomposition lemma 7.1. Chord sequences and bi-universal walks 7.2. Bi-setups and the robust decomposition lemma 8. Proof of Theorem 1.6 9. Proof of Theorem 1.5 References

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1. Introduction The topic of decomposing a graph into a given collection of edge-disjoint subgraphs has a long history. Indeed, in 1892, Walecki [19] proved that every complete graph of odd order has a decomposition into edge-disjoint Hamilton cycles. In a sequence of four papers, we provide a unified approach towards proving three long-standing graph decomposition conjectures for all sufficiently large graphs. 1.1. The 1-factorization conjecture. Vizing’s theorem states that for any graph G of maximum degree ∆, its edge-chromatic number χ′ (G) is either ∆ or ∆ + 1. However, the problem of determining the precise value of χ′ (G) for an arbitrary graph G is NP-complete [8]. Thus, it is of interest to determine classes of graphs G that attain the (trivial) lower bound ∆ – much of the recent book [28] is devoted to the subject. If G is a regular graph then χ′ (G) = ∆(G) precisely when G has a 1factorization: a 1-factorization of a graph G consists of a set of edge-disjoint perfect matchings covering all edges of G. The 1-factorization conjecture states that every regular graph of sufficiently high degree has a 1-factorization. It was first stated explicitly by Chetwynd and Hilton [1, 2] (who also proved partial results). However, they state that according to Dirac, it was already discussed in the 1950s. We prove the 1-factorization conjecture for sufficiently large graphs. Theorem 1.1. There exists an n0 ∈ N such that the following holds. Let n, D ∈ N be such that n ≥ n0 is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a 1-factorization. Equivalently, χ′ (G) = D.

PROOF OF THE 1-FACTORIZATION & HAMILTON DECOMPOSITION CONJECTURES II

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The bound on the minimum degree in Theorem 1.1 is best possible. In fact, a smaller degree bound does not even ensure a single perfect matching. To see this, suppose first that n = 2 (mod 4). Consider the graph which is the disjoint union of two cliques of order n/2 (which is odd). If n = 0 (mod 4), consider the graph obtained from the disjoint union of cliques of orders n/2 − 1 and n/2 + 1 (both odd) by deleting a Hamilton cycle in the larger clique. Perkovic and Reed [26] proved an approximate version of Theorem 1.1 (they assumed that D ≥ n/2 + εn). Recently, this was generalized by Vaughan [29] to multigraphs of bounded multiplicity, thereby proving an approximate version of a ‘multigraph 1-factorization conjecture’ which was raised by Plantholt and Tipnis [27]. Further related results and problems are discussed in the recent monograph [28]. 1.2. The Hamilton decomposition conjecture. A Hamilton decomposition of a graph G consists of a set of edge-disjoint Hamilton cycles covering all the edges of G. A natural extension of this to regular graphs G of odd degree is to ask for a decomposition into Hamilton cycles and one perfect matching (i.e. one perfect matching M in G together with a Hamilton decomposition of G − M ). Nash-Williams [23, 25] raised the problem of finding a Hamilton decomposition in an even-regular graph of sufficiently large degree. The following result completely solves this problem for large graphs. Theorem 1.2. There exists an n0 ∈ N such that the following holds. Let n, D ∈ N be such that n ≥ n0 and D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. The bound on the degree in Theorem 1.2 is best possible (see Proposition 3.1 in [14] for a proof of this). Note that Theorem 1.2 does not quite imply Theorem 1.1, as the degree threshold in the former result is slightly higher. Previous results include the following: Nash-Williams [22] showed that the degree bound in Theorem 1.2 ensures a single Hamilton cycle. Jackson [9] showed that one can ensure close to D/2 − n/6 edge-disjoint Hamilton cycles. More recently, Christofides, K¨ uhn and Osthus [3] obtained an approximate decomposition under the assumption that D ≥ n/2 + εn. Finally, under the same assumption, K¨ uhn and Osthus [16] obtained an exact decomposition (as a consequence of the main result in [15] on Hamilton decompositions of robustly expanding graphs). 1.3. Packing Hamilton cycles in graphs of large minimum degree. Dirac’s theorem is best possible in the sense that one cannot lower the minimum degree condition. Remarkably though, the conclusion can be strengthened considerably: Nash-Williams [24] proved that every graph G on n vertices with minimum degree δ(G) ≥ n/2 contains ⌊5n/224⌋ edge-disjoint Hamilton cycles. Nash-Williams [24, 23, 25] raised the question of finding the best possible bound on the number of edgedisjoint Hamilton cycles in a Dirac graph. This question is answered by Corollary 1.4 below. In fact, we answer a more general form of this question: what is the number of edge-disjoint Hamilton cycles one can guarantee in a graph G of minimum degree δ?

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Let regeven (G) be the largest degree of an even-regular spanning subgraph of G. Then let regeven (n, δ) := min{regeven (G) : |G| = n, δ(G) = δ}. Clearly, in general we cannot guarantee more than regeven (n, δ)/2 edge-disjoint Hamilton cycles in a graph of order n and minimum degree δ. The next result shows that this bound is best possible (if δ < n/2, then regeven (n, δ) = 0). Theorem 1.3. There exists an n0 ∈ N such that the following holds. Suppose that G is a graph on n ≥ n0 vertices with minimum degree δ ≥ n/2. Then G contains at least regeven (n, δ)/2 edge-disjoint Hamilton cycles. K¨ uhn, Lapinskas and Osthus [11] proved Theorem 1.3 in the case when G is not close to one of the extremal graphs for Dirac’s theorem. An approximate version of Theorem 1.3 for δ ≥ n/2 + εn was obtained earlier by Christofides, K¨ uhn and Osthus [3]. Hartke and Seacrest [7] gave a simpler argument with improved error bounds. The following consequence of Theorem 1.3 answers the original question of NashWilliams. Corollary 1.4. There exists an n0 ∈ N such that the following holds. Suppose that G is a graph on n ≥ n0 vertices with minimum degree δ ≥ n/2. Then G contains at least (n − 2)/8 edge-disjoint Hamilton cycles. See [14] for an explanation as to why Corollary 1.4 follows from Theorem 1.3 and for a construction showing the bound on the number of edge-disjoint Hamilton cycles in Corollary 1.4 is best possible (the construction is also described in Section 3.1). 1.4. Overall structure of the argument. For all three of our main results, we split the argument according to the structure of the graph G under consideration: (i) G is close to the complete balanced bipartite graph Kn/2,n/2 ; (ii) G is close to the union of two disjoint copies of a clique Kn/2 ; (iii) G is a ‘robust expander’. Roughly speaking, G is a robust expander if for every set S of vertices, its neighbourhood is at least a little larger than |S|, even if we delete a small proportion of the edges of G. The main result of [15] states that every dense regular robust expander has a Hamilton decomposition. This immediately implies Theorems 1.1 and 1.2 in Case (iii). For Theorem 1.3, Case (iii) is proved in [11] using a more involved argument, but also based on the main result of [15]. Case (ii) is proved in [14, 12]. The current paper is devoted to the proof of Case (i). In [14] we derive Theorems 1.1, 1.2 and 1.3 from the structural results covering Cases (i)–(iii). The arguments in the current paper for Case (i) as well as those in [14] for Case (ii) make use of an ‘approximate’ decomposition result proved in [4]. In both Case (i) and Case (ii) we use the main lemma from [15] (the ‘robust decomposition lemma’) when transforming this approximate decomposition into an exact one.


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1.5. Statement of the main results of this paper. As mentioned above, the focus of this paper is to prove Theorems 1.1, 1.2 and 1.3 when our graph is close to the complete balanced bipartite graph Kn/2,n/2 . More precisely, we say that a graph G on n vertices is ε-bipartite if there is a partition S1 , S2 of V (G) which satisfies the following: • n/2 − 1 < |S1 |, |S2 | < n/2 + 1; • e(S1 ), e(S2 ) ≤ εn2 . The following result implies Theorems 1.1 and 1.2 in the case when our given graph is close to Kn/2,n/2 . Theorem 1.5. There are εex > 0 and n0 ∈ N such that the following holds. Suppose that D ≥ (1/2 − εex )n and D is even and suppose that G is a D-regular graph on n ≥ n0 vertices which is εex -bipartite. Then G has a Hamilton decomposition. The next result implies Theorem 1.3 in the case when our graph is close to Kn/2,n/2 . Theorem 1.6. For each α > 0 there are εex > 0 and n0 ∈ N such that the following holds. Suppose that F is an εex -bipartite graph on n ≥ n0 vertices with δ(F ) ≥ (1/2−εex )n. Suppose that F has a D-regular spanning subgraph G such that n/100 ≤ D ≤ (1/2 − α)n and D is even. Then F contains D/2 edge-disjoint Hamilton cycles. Note that Theorem 1.5 implies that the degree bound in Theorems 1.1 and 1.2 is not tight in the almost bipartite case (indeed, the extremal graph is close to being the union of two cliques). On the other hand, the extremal construction for Corollary 1.4 is close to bipartite (see Section 3.1 for a description). So it turns out that the bound on the number of edge-disjoint Hamilton cycles in Corollary 1.4 is best possible in the almost bipartite case but not when the graph is close to the union of two cliques. In Section 3 we give an outline of the proofs of Theorems 1.5 and 1.6. The results from Sections 4 and 5 are used in both the proofs of Theorems 1.5 and 1.6. In Sections 6 and 7 we build up machinery for the proof of Theorem 1.5. We then prove Theorem 1.6 in Section 8 and Theorem 1.5 in Section 9. 2. Notation and Tools 2.1. Notation. Unless stated otherwise, all the graphs and digraphs considered in this paper are simple and do not contain loops. So in a digraph G, we allow up to two edges between any two vertices; at most one edge in each direction. Given a graph or digraph G, we write V (G) for its vertex set, E(G) for its edge set, e(G) := |E(G)| for the number of its edges and |G| := |V (G)| for the number of its vertices. Suppose that G is an undirected graph. We write δ(G) for the minimum degree of G and ∆(G) for its maximum degree. Given a vertex v of G and a set A ⊆ V (G), we write dG (v, A) for the number of neighbours of v in G which lie in A. Given A, B ⊆ V (G), we write EG (A) for the set of all those edges of G which have both endvertices in A and EG (A, B) for the set of all those edges of G which have one endvertex in A and its other endvertex in B. We also call the edges in EG (A, B) AB-edges of G. We let eG (A) := |EG (A)| and eG (A, B) := |EG (A, B)|. We denote by G[A] the subgraph of G with vertex set A and edge set EG (A). If A ∩ B = ∅,

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we denote by G[A, B] the bipartite subgraph of G with vertex classes A and B and edge set EG (A, B). If A = B we define G[A, B] := G[A]. We often omit the index G if the graph G is clear from the context. A spanning subgraph H of G is an r-factor of G if every vertex has degree r in H. Given a vertex set V and two multigraphs G and H with V (G), V (H) ⊆ V , we write G + H for the multigraph whose vertex set is V (G) ∪ V (H) and in which the multiplicity of xy in G + H is the sum of the multiplicities of xy in G and in H (for all x, y ∈ V (G)∪ V (H)). We say that a graph G has a decomposition into H1 , . . . , Hr if G = H1 + · · · + Hr and the Hi are pairwise edge-disjoint. If G and H are simple graphs, we write G ∪ H for the (simple) graph whose vertex set is V (G) ∪ V (H) and whose edge set is E(G) ∪ E(H). Similarly, G ∩ H denotes the graph whose vertex set is V (G) ∩ V (H) and whose edge set is E(G) ∩ E(H). We write G − H for the subgraph of G which is obtained from G by deleting all the edges in E(G) ∩ E(H). Given A ⊆ V (G), we write G − A for the graph obtained from G by deleting all vertices in A. A path system is a graph Q which is the union of vertex-disjoint paths (some of them might be trivial). We say that P is a path in Q if P is a component of Q and, abusing the notation, sometimes write P ∈ Q for this. If G is a digraph, we write xy for an edge directed from x to y. A digraph G is an oriented graph if there are no x, y ∈ V (G) such that xy, yx ∈ E(G). Unless stated otherwise, when we refer to paths and cycles in digraphs, we mean directed paths and cycles, i.e. the edges on these paths/cycles are oriented consistently. If x is a vertex of a digraph G, then NG+ (x) denotes the outneighbourhood of x, i.e. the set of all those vertices y for which xy ∈ E(G). Similarly, NG− (x) denotes the inneighbourhood of x, i.e. the set of all those vertices y for which yx ∈ E(G). The outdegree of x is − − + d+ G (x) := |NG (x)| and the indegree of x is dG (x) := |NG (x)|. We write δ(G) and ∆(G) for the minimum and maximum degrees of the underlying simple undirected graph of G respectively. For a digraph G, whenever A, B ⊆ V (G) with A∩B = ∅, we denote by G[A, B] the bipartite subdigraph of G with vertex classes A and B whose edges are all the edges of G directed from A to B, and let eG (A, B) denote the number of edges in G[A, B]. We define δ(G[A, B]) to be the minimum degree of the underlying undirected graph of G[A, B] and define ∆(G[A, B]) to be the maximum degree of the underlying undirected graph of G[A, B]. A spanning subdigraph H of G is an r-factor of G if the outdegree and the indegree of every vertex of H is r. If P is a path and x, y ∈ V (P ), we write xP y for the subpath of P whose endvertices are x and y. We define xP y similarly if P is a directed path and x precedes y on P . In order to simplify the presentation, we omit floors and ceilings and treat large numbers as integers whenever this does not affect the argument. The constants in the hierarchies used to state our results have to be chosen from right to left. More precisely, if we claim that a result holds whenever 0 < 1/n ≪ a ≪ b ≪ c ≤ 1 (where n is the order of the graph or digraph), then this means that there are nondecreasing functions f : (0, 1] → (0, 1], g : (0, 1] → (0, 1] and h : (0, 1] → (0, 1] such that the result holds for all 0 < a, b, c ≤ 1 and all n ∈ N with b ≤ f (c), a ≤ g(b)


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and 1/n ≤ h(a). We will not calculate these functions explicitly. Hierarchies with more constants are defined in a similar way. We will write a = b ± c as shorthand for b − c ≤ a ≤ b + c. 2.2. ε-regularity. If G = (A, B) is an undirected bipartite graph with vertex classes A and B, then the density of G is defined as d(A, B) :=

eG (A, B) . |A||B|

For any ε > 0, we say that G is ε-regular if for any A′ ⊆ A and B ′ ⊆ B with |A′ | ≥ ε|A| and |B ′ | ≥ ε|B| we have |d(A′ , B ′ ) − d(A, B)| < ε. We say that G is (ε, ≥ d)-regular if it is ε-regular and has density d′ for some d′ ≥ d − ε. We say that G is [ε, d]-superregular if it is ε-regular and dG (a) = (d ± ε)|B| for every a ∈ A and dG (b) = (d ± ε)|A| for every b ∈ B. G is [ε, ≥ d]-superregular if it is [ε, d′ ]-superregular for some d′ ≥ d. Given disjoint vertex sets X and Y in a digraph G, recall that G[X, Y ] denotes the bipartite subdigraph of G whose vertex classes are X and Y and whose edges are all the edges of G directed from X to Y . We often view G[X, Y ] as an undirected bipartite graph. In particular, we say G[X, Y ] is ε-regular, (ε, ≥ d)-regular, [ε, d]superregular or [ε, ≥ d]-superregular if this holds when G[X, Y ] is viewed as an undirected graph. We often use the following simple proposition which follows easily from the definition of (super-)regularity. We omit the proof, a similar argument can be found e.g. in [15]. Proposition 2.1. Suppose that 0 < 1/m ≪ ε ≤ d′ ≪ d ≤ 1. Let G be a bipartite graph with vertex classes A and B of size m. Suppose that G′ is obtained from G by removing at most d′ m vertices from each vertex class and at most d′ m edges incident √ to each vertex from G. If G is [ε, d]-superregular then G′ is [2 d′ , d]-superregular. We will also use the following simple fact. Fact 2.2. Let ε > 0. Suppose that G is a√bipartite graph with vertex classes of size n such that δ(G) ≥ (1 − ε)n. Then G is [ ε, 1]-superregular. 2.3. A Chernoff-Hoeffding bound. We will often use the following ChernoffHoeffding bound for binomial and hypergeometric distributions (see e.g. [10, Corollary 2.3 and Theorem 2.10]). Recall that the binomial random variable with parameters (n, p) is the sum of n independent Bernoulli variables, each taking value 1 with probability p or 0 with probability 1 − p. The hypergeometric random variable X with parameters (n, m, k) is defined as follows. We let N be a set of size n, fix S ⊆ N of size |S| = m, pick a uniformly random T ⊆ N of size |T | = k, then define X := |T ∩ S|. Note that EX = km/n. Proposition 2.3. Suppose X has binomial or hypergeometric distribution and 0 < a < 3/2. Then P(|X − EX| ≥ aEX) ≤ 2e−

a2 EX 3

.

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3. Overview of the proofs of Theorems 1.5 and 1.6 Note that, unlike in Theorem 1.5, in Theorem 1.6 we do not require a complete decomposition of our graph F into edge-disjoint Hamilton cycles. Therefore, the proof of Theorem 1.5 is considerably more involved than the proof of Theorem 1.6. Moreover, the ideas in the proof of Theorem 1.6 are all used in the proof of Theorem 1.5 too. 3.1. Proof overview for Theorem 1.6. Let F be a graph on n vertices with δ(F ) ≥ (1/2−o(1))n which is close to the balanced bipartite graph Kn/2,n/2 . Further, suppose that G is a D-regular spanning subgraph of F as in Theorem 1.6. Then there is a partition A, B of V (F ) such that A and B are of roughly equal size and most edges in F go between A and B. Our ultimate aim is to construct D/2 edge-disjoint Hamilton cycles in F . Suppose first that, in the graph F , both A and B are independent sets of equal size. So F is an almost complete balanced bipartite graph. In this case, the densest spanning even-regular subgraph G of F is also almost complete bipartite. This means that one can extend existing techniques (developed e.g. in [3, 5, 6, 7, 21]) to find an approximate Hamilton decomposition. This is achieved in [4] and is more than enough to prove Theorem 1.6 in this case. (We state the main result from [4] as Lemma 8.1 in the current paper.) The real difficulties arise when (i) F is unbalanced; (ii) F has vertices having high degree in both A and B (these are called exceptional vertices). To illustrate (i), consider the following example due to Babai (which is the extremal construction for Corollary 1.4). Consider the graph F on n = 8k + 2 vertices consisting of one vertex class A of size 4k + 2 containing a perfect matching and no other edges, one empty vertex class B of size 4k, and all possible edges between A and B. Thus the minimum degree of F is 4k + 1 = n/2. Then one can use Tutte’s factor theorem to show that the largest even-regular spanning subgraph G of F has degree D = 2k = (n − 2)/4. Note that to prove Theorem 1.6 in this case, each of the D/2 = k Hamilton cycles we find must contain exactly two of the 2k + 1 edges in A. In this way, we can ‘balance out’ the difference in the vertex class sizes. More generally we will construct our Hamilton cycles in two steps. In the first step, we find a path system J which balances out the vertex class sizes (so in the above example, J would contain two edges in A). Then we extend J into a Hamilton cycle using only AB-edges in F . It turns out that the first step is the difficult one. It is easy to see that a path system J will balance out the sizes of A and B (in the sense that the number of uncovered vertices in A and B is the same) if and only if (3.1)

eJ (A) − eJ (B) = |A| − |B|.

Note that any Hamilton cycle also satisfies this identity. So we need to find a set of D/2 path systems J satisfying (3.1) (where D is the degree of G). This is achieved (amongst other things) in Sections 5.2 and 5.3. As indicated above, our aim is to use Lemma 8.1 in order to extend each such J into a Hamilton cycle. To apply Lemma 8.1 we also need to extend the balancing


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path systems J into ‘balanced exceptional (path) systems’ which contain all the exceptional vertices from (ii). This is achieved in Section 5.4. Lemma 8.1 also assumes that the path systems are ‘localized’ with respect to a given subpartition of A, B (i.e. they are induced by a small number of partition classes). Section 5.1 prepares the ground for this. Finding the balanced exceptional systems is extremely difficult if G contains edges between the set A0 of exceptional vertices in A and the set B0 of exceptional vertices in B. So in a preliminary step, we find and remove a small number of edge-disjoint Hamilton cycles covering all A0 B0 -edges in Section 4. We put all these steps together in Section 8. (Sections 6, 7 and 9 are only relevant for the proof of Theorem 1.5.) 3.2. Proof overview for Theorem 1.5. The main result of this paper is Theorem 1.5. Suppose that G is a D-regular graph satisfying the conditions of that theorem. Using the approach of the previous subsection, one can obtain an approximate decomposition of G, i.e. a set of edge-disjoint Hamilton cycles covering almost all edges of G. However, one does not have any control over the ‘leftover’ graph H, which makes a complete decomposition seem infeasible. This problem was overcome in [15] by introducing the concept of a ‘robustly decomposable graph’ Grob . Roughly speaking, this is a sparse regular graph with the following property: given any very sparse regular graph H with V (H) = V (Grob ) which is edge-disjoint from Grob , one can guarantee that Grob ∪ H has a Hamilton decomposition. This leads to the following strategy to obtain a decomposition of G: (1) find a (sparse) robustly decomposable graph Grob in G and let G′ denote the leftover; (2) find an approximate Hamilton decomposition of G′ and let H denote the (very sparse) leftover; (3) find a Hamilton decomposition of Grob ∪ H. It is of course far from obvious that such a graph Grob exists. By assumption our graph G can be partitioned into two classes A and B of almost equal size such that almost all the edges in G go between A and B. If both A and B are independent sets of equal size then the ‘robust decomposition lemma’ of [15] guarantees our desired subgraph Grob of G. Of course, in general our graph G will contain edges in A and B. Our aim is therefore to replace such edges with ‘fictive edges’ between A and B, so that we can apply the robust decomposition lemma (which is introduced in Section 7). More precisely, similarly as in the proof of Theorem 1.6, we construct a collection of localized balanced exceptional systems. Together these path systems contain all the edges in G[A] and G[B]. Again, each balanced exceptional system balances out the sizes of A and B and covers the exceptional vertices in G (i.e. those vertices having high degree into both A and B). By replacing edges of the balanced exceptional systems with fictive edges, we obtain from G an auxiliary (multi)graph G∗ which only contains edges between A and B and which does not contain the exceptional vertices of G. This will allow us to apply the robust decomposition lemma. In particular this ensures that each Hamilton cycle obtained in G∗ contains a collection of fictive edges corresponding to

´ ¨ 10 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN

a single balanced exceptional system (the set-up of the robust decomposition lemma does allow for this). Each such Hamilton cycle in G∗ then corresponds to a Hamilton cycle in G. We now give an example of how we introduce fictive edges. Let m be an integer so that (m − 1)/2 is even. Set m′ := (m − 1)/2 and m′′ := (m + 1)/2. Define the graph G as follows: Let A and B be disjoint vertex sets of size m. Let A1 , A2 be a partition of A and B1 , B2 be a partition of B such that |A1 | = |B1 | = m′′ . Add all edges between A and B. Add a matching M1 = {e1 , . . . , em′ /2 } covering precisely the vertices of A2 and add a matching M2 = {e′1 , . . . , e′m′ /2 } covering precisely the vertices of B2 . Finally add a vertex v which sends an edge to every vertex in A1 ∪B1 . So G is (m + 1)-regular (and v would be regarded as a exceptional vertex). Now pair up each edge ei with the edge e′i . Write ei = x2i−1 x2i and e′i = y2i−1 y2i for each 1 ≤ i ≤ m′ /2. Let A1 = {a1 , . . . , am′′ } and B1 = {b1 , . . . , bm′′ } and write fi := ai bi for all 1 ≤ i ≤ m′′ . Obtain G∗ from G by deleting v together with the edges in M1 ∪ M2 and by adding the following fictive edges: add fi for each 1 ≤ i ≤ m′′ and add xj yj for each 1 ≤ j ≤ m′ . Then G∗ is a balanced bipartite (m + 1)-regular multigraph containing only edges between A and B. First, note that any Hamilton cycle C ∗ in G∗ that contains precisely one fictive edge fi for some 1 ≤ i ≤ m′′ corresponds to a Hamilton cycle C in G, where we replace the fictive edge fi with ai v and bi v. Next, consider any Hamilton cycle C ∗ in G∗ that contains precisely three fictive edges; fi for some 1 ≤ i ≤ m′′ together with x2j−1 y2j−1 and x2j y2j for some 1 ≤ j ≤ m′ /2. Further suppose C ∗ traverses the vertices ai , bi , x2j−1 , y2j−1 , x2j , y2j in this order. Then C ∗ corresponds to a Hamilton cycle C in G, where we replace the fictive edges with ai v, bi v, ej and e′j (see Figure 1). Here the path system J formed by the edges ai v, bi v, ej and e′j is an example of a balanced exceptional system. The above ideas are formalized in Section 6. v

fi

ai

bi

y2j

x2j

x2j−1

A

y2j−1

B

Figure 1. Transforming the problem of finding a Hamilton cycle in G into finding a Hamilton cycle in the balanced bipartite graph G∗

PROOF OF THE 1-FACTORIZATION & HAMILTON DECOMPOSITION CONJECTURES II 11

We can now summarize the steps leading to proof of Theorem 1.5. In Section 4, we find and remove a set of edge-disjoint Hamilton cycles covering all edges in G[A0 , B0 ]. We can then find the localized balanced exceptional systems in Section 5. After this, we need to extend and combine them into certain path systems and factors (which contain fictive edges) in Section 6, before we can use them as an ‘input’ for the robust decomposition lemma in Section 7. Finally, all these steps are combined in Section 9 to prove Theorem 1.5. 4. Eliminating edges between the exceptional sets Suppose that G is a D-regular graph as in Theorem 1.5. The purpose of this section is to prove Corollary 4.13. Roughly speaking, given K ∈ N, this corollary states that one can delete a small number of edge-disjoint Hamilton cycles from G to obtain a spanning subgraph G′ of G and a partition A, A0 , B, B0 of V (G) such that (amongst others) the following properties hold: • almost all edges of G′ join A ∪ A0 to B ∪ B0 ; • |A| = |B| is divisible by K; • every vertex in A has almost all its neighbours in B ∪ B0 and every vertex in B has almost all its neighbours in A ∪ A0 ; • A0 ∪ B0 is small and there are no edges between A0 and B0 in G′ .

We will call (G′ , A, A0 , B, B0 ) a framework. (The formal definition of a framework is stated before Lemma 4.12.) Both A and B will then be split into K clusters of equal size. Our assumption that G is εex -bipartite easily implies that there is such a partition A, A0 , B, B0 which satisfies all these properties apart from the property that there are no edges between A0 and B0 . So the main part of this section shows that we can cover the collection of all edges between A0 and B0 by a small number of edge-disjoint Hamilton cycles. Since Corollary 4.13 will also be used in the proof of Theorem 1.6, instead of working with regular graphs we need to consider so-called balanced graphs. We also need to find the above Hamilton cycles in the graph F ⊇ G rather than in G itself (in the proof of Theorem 1.5 we will take F to be equal to G). More precisely, suppose that G is a graph and that A′ , B ′ is a partition of V (G), where A′ = A0 ∪ A, B ′ = B0 ∪ B and A, A0 , B, B0 are disjoint. Then we say that G is D-balanced (with respect to (A, A0 , B, B0 )) if (B1) eG (A′ ) − eG (B ′ ) = (|A′ | − |B ′ |)D/2; (B2) all vertices in A0 ∪ B0 have degree exactly D.

Proposition 4.1 below implies that whenever A, A0 , B, B0 is a partition of the vertex set of a D-regular graph H, then H is D-balanced with respect to (A, A0 , B, B0 ). Moreover, note that if G is DG -balanced with respect to (A, A0 , B, B0 ) and H is a spanning subgraph of G which is DH -balanced with respect to (A, A0 , B, B0 ), then G − H is (DG − DH )-balanced with respect to (A, A0 , B, B0 ). Furthermore, a graph G is D-balanced with respect to (A, A0 , B, B0 ) if and only if G is D-balanced with respect to (B, B0 , A, A0 ).


Proposition 4.1. Let H be a graph and let A′ , B ′ be a partition of V (H). Suppose that A0 , A is a partition of A′ and that B0 , B is a partition of B ′ such that |A| = |B|. Suppose that dH (v) = D for every v ∈ A0 ∪ B0 and dH (v) = D ′ for every v ∈ A ∪ B. Then eH (A′ ) − eH (B ′ ) = (|A′ | − |B ′ |)D/2. Proof. Note that X

dH (x, B ′ ) = eH (A′ , B ′ ) =

x∈A′

X

dH (y, A′ ).

y∈B ′

Moreover, 2eH (A′ ) =

dH (x, B ′ )

x∈A

X

x∈A′

X

X

dH (y, A′ ).

X

(D−dH (x, B ′ ))+

X

(D−dH (y, A′ ))+

x∈A0

X

(D ′ −dH (x, B ′ )) = D|A0 |+D ′ |A|−

and 2eH (B ′ ) =

y∈B0

y∈B

(D ′ −dH (y, A′ )) = D|B0 |+D ′ |B|−

y∈B ′

Therefore 2eH (A′ )−2eH (B ′ ) = D(|A0 |−|B0 |)+D ′ (|A|−|B|) = D(|A0 |−|B0 |) = D(|A′ |−|B ′ |), as desired.

The following observation states that balancedness is preserved under suitable modifications of the partition. Proposition 4.2. Let H be D-balanced with respect to (A, A0 , B, B0 ). Suppose that A′0 , B0′ is a partition of A0 ∪B0 . Then H is D-balanced with respect to (A, A′0 , B, B0′ ). Proof. Observe that the general result follows if we can show that H is D-balanced with respect to (A, A′0 , B, B0′ ), where A′0 = A0 ∪ {v}, B0′ = B0 \{v} and v ∈ B0 . (B2) is trivially satisfied in this case, so we only need to check (B1) for the new partition. For this, let A′ := A0 ∪ A and B ′ := B0 ∪ B. Now note that (B1) for the original partition implies that eH (A′0 ∪ A) − eH (B0′ ∪ B) = eH (A′ ) + dH (v, A′ ) − (eH (B ′ ) − dH (v, B ′ ))

= (|A′ | − |B ′ |)D/2 + D = (|A′0 ∪ A| − |B0′ ∪ B|)D/2.

Thus (B1) holds for the new partition.

Suppose that G is a graph and A′ , B ′ is a partition of V (G). For every vertex v ∈ A′ we call dG (v, A′ ) the internal degree of v in G. Similarly, for every vertex v ∈ B ′ we call dG (v, B ′ ) the internal degree of v in G. Given a graph F and a spanning subgraph G of F , we say that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-weak framework if the following holds, where A′ := A0 ∪ A, B ′ := B0 ∪ B and n := |G| = |F |: (WF1) A, A0 , B, B0 forms a partition of V (G) = V (F ); (WF2) G is D-balanced with respect to (A, A0 , B, B0 ); (WF3) eG (A′ ), eG (B ′ ) ≤ εn2 ;


(WF4) |A| = |B| is divisible by K. Moreover, a + b ≤ εn, where a := |A0 | and b := |B0 |; (WF5) all vertices in A ∪ B have internal degree at most ε′ n in F ; (WF6) any vertex v has internal degree at most dG (v)/2 in G. Throughout the paper, when referring to internal degrees without mentioning the partition, we always mean with respect to the partition A′ , B ′ , where A′ = A0 ∪ A and B ′ = B0 ∪ B. Moreover, a and b will always denote |A0 | and |B0 |. We say that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-pre-framework if it satisfies (WF1)–(WF5). The following observation states that pre-frameworks are preserved if we remove suitable balanced subgraphs. Proposition 4.3. Let ε, ε′ > 0 and K, DG , DH ∈ N. Let (F, G, A, A0 , B, B0 ) be an (ε, ε′ , K, DG )-pre framework. Suppose that H is a DH -regular spanning subgraph of F such that G∩H is DH -balanced with respect to (A, A0 , B, B0 ). Let F ′ := F −H and G′ := G − H. Then (F ′ , G′ , A, A0 , B, B0 ) is an (ε, ε′ , K, DG − DH )-pre framework.

Proof. Note that all required properties except possibly (WF2) are not affected by removing edges. But G′ satisfies (WF2) since G ∩ H is DH -balanced with respect to (A, A0 , B, B0 ). Lemma 4.4. Let 0 < 1/n ≪ ε ≪ ε′ , 1/K ≪ 1 and let D ≥ n/200. Suppose that F is a graph on n vertices which is ε-bipartite and that G is a D-regular spanning subgraph of F . Then there is a partition A, A0 , B, B0 of V (G) = V (F ) so that (F, G, A, A0 , B, B0 ) is an (ε1/3 , ε′ , K, D)-weak framework.

Proof. Let S1 , S2 be a partition of V (F ) which is guaranteed by the assumption √ that F is ε-bipartite. Let S be the set of all those vertices x √ ∈ S1 with dF (x, S1 ) ≥ εn together with all those√vertices x ∈ S2 with dF (x, S2 ) ≥ εn. Since F is ε-bipartite, it follows that |S| ≤ 4 εn. Given a partition X, Y of V (F ), we say that v ∈ X is bad for X, Y if dG (v, X) > dG (v, Y ) and similarly that v ∈ Y is bad for X, Y if dG (v, Y ) > dG (v, X). Suppose that there is a vertex v ∈ S which is bad for S1 , S2 . Then we move v into the class which does not currently contain v to obtain a new partition S1′ , S2′ . We do not change the set S. If there is a vertex v ′ ∈ S which is bad for S1′ , S2′ , then again we move it into the other class. We repeat this process. After each step, the number of edges in G between the two classes increases, so this process has to terminate with some partition A′ , B ′ such that A′ △ S1 ⊆ S and B ′ △ S2 ⊆ S. Clearly, no vertex in S is now bad for A′ , B ′ . Also, for any v ∈ A′ \ S we have √ √ (4.1) dG (v, A′ ) ≤ dF (v, A′ ) ≤ dF (v, S1 ) + |S| ≤ εn + 4 εn < ε′ n < D/2 = dG (v)/2.

Similarly, dG (v, B ′ ) < ε′ n < dG (v)/2 for all v ∈ B ′ \ S. Altogether this implies that no vertex is bad for A′ , B ′ and thus (WF6) holds. Also note that eG (A′ , B ′ ) ≥ eG (S1 , S2 ) ≥ e(G) − 2εn2 . So (4.2)

eG (A′ ), eG (B ′ ) ≤ 2εn2 .


This implies (WF3). Without loss of generality we may assume that |A′ | ≥ |B ′ |. Let A′0 denote the set of all those vertices v ∈ A′ for which dF (v, A′ ) ≥ ε′ n. Define B0′ ⊆ B ′ similarly. We will choose sets A ⊆ A′ \ A′0 and A0 ⊇ A′0 and sets B ⊆ B ′ \ B0′ and B0 ⊇ B0′ such that |A| = |B| is divisible by K and so that A, A0 and B, B0 are partitions of A′ and B ′ respectively. We obtain such sets by moving at most ||A′ \ A′0 | − |B ′ \ B0′ || + K vertices from A′ \ A′0 to A′0 and at most ||A′ \ A′0 | − |B ′ \ B0′ || + K vertices from B ′ \ B0′ to B0′ . The choice of A, A0 , B, B0 is such that (WF1) and (WF5) hold. Further, since |A| = |B|, Proposition 4.1 implies (WF2). In order to verify (WF4), it remains to show that a + b = |A0 ∪ B0 | ≤ ε1/3 n. But ′ ′ ′ (4.1) together with its analogue √ for the vertices in B \ S implies that A0 ∪ B0 ⊆ S. ′ ′ Thus |A0 | + |B0 | ≤ |S| ≤ 4 εn. Moreover, (WF2), (4.2) and our assumption that D ≥ n/200 together imply that |A′ | − |B ′ | = (eG (A′ ) − eG (B ′ ))/(D/2) ≤ 2εn2 /(D/2) ≤ 800εn.

So altogether, we have

a + b ≤ |A′0 ∪ B0′ | + 2 |A′ \ A′0 | − |B ′ \ B0′ | + 2K √ ≤ 4 εn + 2 |A′ | − |B ′ | − (|A′0 | − |B0′ |) + 2K √ √ ≤ 4 εn + 1600εn + 8 εn + 2K ≤ ε1/3 n. Thus (WF4) holds.

Throughout this and the next section, we will often use the following result, which is a simple consequence of Vizing’s theorem and was first observed by McDiarmid and independently by de Werra (see e.g. [30]). Proposition 4.5. Let H be a graph with maximum degree at most ∆. Then E(H) can be decomposed into ∆+1 edge-disjoint matchings M1 , . . . , M∆+1 such that ||Mi |− |Mj || ≤ 1 for all i, j ≤ ∆ + 1. Our next goal is to cover the edges of G[A0 , B0 ] by edge-disjoint Hamilton cycles. To do this, we will first decompose G[A0 , B0 ] into a collection of matchings. We will then extend each such matching into a system of vertex-disjoint paths such that altogether these paths cover every vertex in G[A0 , B0 ], each path has its endvertices in A ∪ B and the path system is 2-balanced. Since our path system will only contain a small number of nontrivial paths, we can then extend the path system into a Hamilton cycle (see Lemma 4.10). We will call the path systems we are working with A0 B0 -path systems. More precisely, an A0 B0 -path system (with respect to (A, A0 , B, B0 )) is a path system Q satisfying the following properties: • Every vertex in A0 ∪ B0 is an internal vertex of a path in Q. • A ∪ B contains the endpoints of each path in Q but no internal vertex of a path in Q. The following observation (which motivates the use of the word ‘balanced’) will often be helpful.


Proposition 4.6. Let A0 , A, B0 , B be a partition of a vertex set V . Then an A0 B0 path system Q with V (Q) ⊆ V is 2-balanced with respect to (A, A0 , B, B0 ) if and only if the number of vertices in A which are endpoints of nontrivial paths in Q equals the number of vertices in B which are endpoints of nontrivial paths in Q. Proof. Note that by definition any A0 B0 -path system satisfies (B2), so we only need to consider (B1). Let nA be the number of vertices in A which are endpoints of nontrivial paths in Q and define nB similarly. Let a := |A0 |, b := |B0 |, A′ := A ∪ A0 and B ′ := B ∪ B0 . Since dQ (v) = 2 for all v ∈ A0 and since every vertex in A is either an endpoint of a nontrivial path in Q or has degree zero in Q, we have X 2eQ (A′ ) + eQ (A′ , B ′ ) = dQ (v) = 2a + nA . v∈A′

So nA = 2(eQ (A′ ) − a) + eQ (A′ , B ′ ), and similarly nB = 2(eQ (B ′ ) − b) + eQ (A′ , B ′ ). Therefore, nA = nB if and only if 2(eQ (A′ ) − eQ (B ′ ) − a + b) = 0 if and only if Q satisfies (B1), as desired. The next observation shows that if we have a suitable path system satisfying (B1), we can extend it into a path system which also satisfies (B2). Lemma 4.7. Let 0 < 1/n ≪ α ≪ 1. Let G be a graph on n vertices such that there is a partition A′ , B ′ of V (G) which satisfies the following properties: (i) A′ = A0 ∪ A, B ′ = B0 ∪ B and A0 , A, B0 , B are disjoint; (ii) |A| = |B| and a + b ≤ αn, where a := |A0 | and b := |B0 |; (iii) if v ∈ A0 then dG (v, B) ≥ 4αn and if v ∈ B0 then dG (v, A) ≥ 4αn.

Let Q′ ⊆ G be a path system consisting of at most αn nontrivial paths such that A∪B contains no internal vertex of a path in Q′ and eQ′ (A′ ) − eQ′ (B ′ ) = a − b. Then G contains a 2-balanced A0 B0 -path system Q (with respect to (A, A0 , B, B0 )) which extends Q′ and consists of at most 2αn nontrivial paths. Furthermore, E(Q) \ E(Q′ ) consists of A0 B- and AB0 -edges only. Proof. Since A ∪ B contains no internal vertex of a path in Q′ and since Q′ contains at most αn nontrivial paths, it follows that at most 2αn vertices in A ∪ B lie on nontrivial paths in Q′ . We will now extend Q′ into an A0 B0 -path system Q consisting of at most a + b + αn ≤ 2αn nontrivial paths as follows: • for every vertex v ∈ A0 , we join v to 2 − dQ′ (v) vertices in B; • for every vertex v ∈ B0 , we join v to 2 − dQ′ (v) vertices in A.

Condition (iii) and the fact that at most 2αn vertices in A ∪ B lie on nontrivial paths in Q′ together ensure that we can extend Q′ in such a way that the endvertices in A ∪ B are distinct for different paths in Q. Note that eQ (A′ ) − eQ (B ′ ) = eQ′ (A′ ) − eQ′ (B ′ ) = a − b. Therefore, Q is 2-balanced with respect to (A, A0 , B, B0 ). The next lemma constructs a small number of 2-balanced A0 B0 -path systems covering the edges of G[A0 , B0 ]. Each of these path systems will later be extended into a Hamilton cycle.


Lemma 4.8. Let 0 < 1/n ≪ ε ≪ ε′ , 1/K ≪ α ≪ 1. Let F be a graph on n vertices and let G be a spanning subgraph of F . Suppose that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-weak framework with δ(F ) ≥ (1/4 + α)n and D ≥ n/200. Then for some r ∗ ≤ εn the graph G contains r ∗ edge-disjoint 2-balanced A0 B0 -path systems Q1 , . . . , Qr∗ which satisfy the following properties: (i) Together Q1 , . . . , Qr∗ cover all edges in G[A0 , B0 ]; (ii) For each i ≤ r ∗ , Qi contains at most 2εn nontrivial paths; (iii) For each i ≤ r ∗ , Qi does not contain any edge from G[A, B]. Proof. (WF4) implies that |A0 | + |B0 | ≤ εn. Thus, by Proposition 4.5, there exists a collection M1′ , . . . , Mr′ ∗ of r ∗ edge-disjoint matchings in G[A0 , B0 ] that together cover all the edges in G[A0 , B0 ], where r ∗ ≤ εn. We may assume that a ≥ b (the case when b > a follows analogously). We will use edges in G[A′ ] to extend each Mi′ into a 2-balanced A0 B0 -path system. (WF2) implies that eG (A′ ) ≥ (a − b)D/2. Since dG (v) = D for all v ∈ A0 ∪ B0 by (WF2), (WF5) and (WF6) imply that ∆(G[A′ ]) ≤ D/2. Thus Proposition 4.5 implies that E(G[A′ ]) can be decomposed into ⌊D/2⌋ + 1 edge-disjoint matchings MA,1 , . . . , MA,⌊D/2⌋+1 such that ||MA,i | − |MA,j || ≤ 1 for all i, j ≤ ⌊D/2⌋ + 1. Notice that at least εn of the matchings MA,i are such that |MA,i | ≥ a− b. Indeed, otherwise we have that (a − b)D/2 ≤ eG (A′ ) ≤ εn(a − b) + (a − b − 1)(D/2 + 1 − εn) = (a − b)D/2 + a − b − D/2 − 1 + εn

< (a − b)D/2 + 2εn − D/2 < (a − b)D/2, a contradiction. (The last inequality follows since D ≥ n/200.) In particular, this implies that G[A′ ] contains r ∗ edge-disjoint matchings M1′′ , . . . , Mr′′∗ that each consist of precisely a − b edges. For each i ≤ r ∗ , set Mi := Mi′ ∪ Mi′′ . So for each i ≤ r ∗ , Mi is a path system consisting of at most b + (a − b) = a ≤ εn nontrivial paths such that A ∪ B contains no internal vertex of a path in Mi and eMi (A′ ) − eMi (B ′ ) = eMi′′ (A′ ) = a − b. Suppose for some 0 ≤ r < r ∗ we have already found a collection Q1 , . . . , Qr of r edge-disjoint 2-balanced A0 B0 -path systems which satisfy the following properties for each i ≤ r: (α)i (β)i (γ)i (δ)i

Qi contains at most 2εn nontrivial paths; Mi ⊆ Qi ; Qi and Mj are edge-disjoint for each j ≤ r ∗ such that i 6= j; Qi contains no edge from G[A, B].

(Note that (α)0 –(δ)0 are vacuously true.) Let G′ denote the spanning subgraph of G obtained from G by deleting the edges lying in Q1 ∪ · · · ∪ Qr . (WF2), (WF4) and (WF6) imply that, if v ∈ A0 , dG′ (v, B) ≥ D/2 − εn − 2r ≥ 4εn and if v ∈ B0 then dG′ (v, A) ≥ 4εn. Thus Lemma 4.7 implies that G′ contains a 2-balanced A0 B0 -path system Qr+1 that satisfies (α)r+1 –(δ)r+1 .


So we can proceed in this way in order to obtain edge-disjoint 2-balanced A0 B0 path systems Q1 , . . . , Qr∗ in G such that (α)i –(δ)i hold for each i ≤ r ∗ . Note that (i)–(iii) follow immediately from these conditions, as desired. The next lemma (Corollary 5.4 in [13]) allows us to extend a 2-balanced path system into a Hamilton cycle. Corollary 5.4 concerns so-called ‘(A, B)-balanced’path systems rather than 2-balanced A0 B0 -path systems. But the latter satisfies the requirements of the former by Proposition 4.6. Lemma 4.9. Let 0 < 1/n ≪ ε′ ≪ α ≪ 1. Let F be a graph and suppose that A0 , A, B0 , B is a partition of V (F ) such that |A| = |B| = n. Let H be a bipartite subgraph of F with vertex classes A and B such that δ(H) ≥ (1/2 + α)n. Suppose that Q is a 2-balanced A0 B0 -path system with respect to (A, A0 , B, B0 ) in F which consists of at most ε′ n nontrivial paths. Then F contains a Hamilton cycle C which satisfies the following properties: • Q ⊆ C; • E(C) \ E(Q) consists of edges from H. Now we can apply Lemma 4.9 to extend a 2-balanced A0 B0 -path system in a pre-framework into a Hamilton cycle. Lemma 4.10. Let 0 < 1/n ≪ ε ≪ ε′ , 1/K ≪ α ≪ 1. Let F be a graph on n vertices and let G be a spanning subgraph of F . Suppose that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-pre-framework, i.e. it satisfies (WF1)–(WF5). Suppose also that δ(F ) ≥ (1/4 + α)n. Let Q be a 2-balanced A0 B0 -path system with respect to (A, A0 , B, B0 ) in G which consists of at most ε′ n nontrivial paths. Then F contains a Hamilton cycle C which satisfies the following properties: (i) Q ⊆ C; (ii) E(C) \ E(Q) consists of AB-edges; (iii) C ∩ G is 2-balanced with respect to (A, A0 , B, B0 ). Proof. Note that (WF4), (WF5) and our assumption that δ(F ) ≥ (1/4 + α)n together imply that every vertex x ∈ A satisfies dF (x, B) ≥ dF (x, B ′ ) − |B0 | ≥ dF (x) − ε′ n − |B0 | ≥ (1/4 + α/2)n ≥ (1/2 + α/2)|B|.

Similarly, dF (x, A) ≥ (1/2 + α/2)|A| for all x ∈ B. Thus, δ(F [A, B]) ≥ (1/2 + α/2)|A|. Applying Lemma 4.9 with F [A, B] playing the role of H, we obtain a Hamilton cycle C in F that satisfies (i) and (ii). To verify (iii), note that (ii) and the 2-balancedness of Q together imply that eC∩G (A′ ) − eC∩G (B ′ ) = eQ (A′ ) − eQ (B ′ ) = a − b. Since every vertex v ∈ A0 ∪ B0 satisfies dC∩G (v) = dQ (v) = 2, (iii) holds.

We now combine Lemmas 4.8 and 4.10 to find a collection of edge-disjoint Hamilton cycles covering all the edges in G[A0 , B0 ].


Lemma 4.11. Let 0 < 1/n ≪ ε ≪ ε′ , 1/K ≪ α ≪ 1 and let D ≥ n/100. Let F be a graph on n vertices and let G be a spanning subgraph of F . Suppose that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-weak framework with δ(F ) ≥ (1/4+α)n. Then for some r ∗ ≤ εn the graph F contains edge-disjoint Hamilton cycles C1 , . . . , Cr∗ which satisfy the following properties: (i) Together C1 , . . . , Cr∗ cover all edges in G[A0 , B0 ]; (ii) (C1 ∪ · · · ∪ Cr∗ ) ∩ G is 2r ∗ -balanced with respect to (A, A0 , B, B0 ).

Proof. Apply Lemma 4.8 to obtain a collection of r ∗ ≤ εn edge-disjoint 2-balanced A0 B0 -path systems Q1 , . . . , Qr∗ in G which satisfy Lemma 4.8(i)–(iii). We will extend each Qi to a Hamilton cycle Ci . Suppose that for some 0 ≤ r < r ∗ we have found a collection C1 , . . . , Cr of r edge-disjoint Hamilton cycles in F such that the following holds for each 0 ≤ i ≤ r: (α)i Qi ⊆ Ci ; (β)i E(Ci ) \ E(Qi ) consists of AB-edges; (γ)i G ∩ Ci is 2-balanced with respect to (A, A0 , B, B0 ). (Note that (α)0 –(γ)0 are vacuously true.) Let Hr := C1 ∪ · · · ∪ Cr (where H0 := (V (G), ∅)). So Hr is 2r-regular. Further, since G ∩ Ci is 2-balanced for each i ≤ r, G∩Hr is 2r-balanced. Let Gr := G−Hr and Fr := F −Hr . Since (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-pre-framework, Proposition 4.3 implies that (Fr , Gr , A, A0 , B, B0 ) is an (ε, ε′ , K, D − 2r)-pre-framework. Moreover, δ(Fr ) ≥ δ(F ) − 2r ≥ (1/4 + α/2)n. Lemma 4.8(iii) and (β)1 –(β)r together imply that Qr+1 lies in Gr . Therefore, Lemma 4.10 implies that Fr contains a Hamilton cycle Cr+1 which satisfies (α)r+1 – (γ)r+1 . So we can proceed in this way in order to obtain r ∗ edge-disjoint Hamilton cycles C1 , . . . , Cr∗ in F such that for each i ≤ r ∗ , (α)i –(γ)i hold. Note that this implies that (ii) is satisfied. Further, the choice of Q1 , . . . , Qr∗ ensures that (i) holds. Given a graph G, we say that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework if the following holds, where A′ := A0 ∪ A, B ′ := B0 ∪ B and n := |G|: (FR1) A, A0 , B, B0 forms a partition of V (G); (FR2) G is D-balanced with respect to (A, A0 , B, B0 ); (FR3) eG (A′ ), eG (B ′ ) ≤ εn2 ; (FR4) |A| = |B| is divisible by K. Moreover, b ≤ a and a + b ≤ εn, where a := |A0 | and b := |B0 |; (FR5) all vertices in A ∪ B have internal degree at most ε′ n in G; (FR6) e(G[A0 , B0 ]) = 0; (FR7) all vertices v ∈ V (G) have internal degree at most dG (v)/2 + εn in G. Note that the main differences to a weak framework are (FR6) and the fact that a weak framework involves an additional graph F . In particular (FR1)–(FR4) imply (WF1)–(WF4). Suppose that ε1 ≥ ε, ε′1 ≥ ε′ and that K1 divides K. Then note that every (ε, ε′ , K, D)-framework is also an (ε1 , ε′1 , K1 , D)-framework. Lemma 4.12. Let 0 < 1/n ≪ ε ≪ ε′ , 1/K ≪ α ≪ 1 and let D ≥ n/100. Let F be a graph on n vertices and let G be a spanning subgraph of F . Suppose that


(F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-weak framework. Suppose also that δ(F ) ≥ (1/4 + α)n and |A0 | ≥ |B0 |. Then the following properties hold: (i) there is an (ε, ε′ , K, DG′ )-framework (G′ , A, A0 , B, B0 ) such that G′ is a spanning subgraph of G with DG′ ≥ D − 2εn; (ii) there is a set of (D − DG′ )/2 ≤ εn edge-disjoint Hamilton cycles in F − G′ containing all edges of G − G′ . In particular, if D is even then DG′ is even.

Proof. Lemma 4.11 implies that there exists some r ∗ ≤ εn such that F contains a spanning subgraph H satisfying the following properties: (a) H is 2r ∗ -regular; (b) H contains all the edges in G[A0 , B0 ]; (c) G ∩ H is 2r ∗ -balanced with respect to (A, A0 , B, B0 ); (d) H has a decomposition into r ∗ edge-disjoint Hamilton cycles. Set G′ := G − H. Then (G′ , A, A0 , B, B0 ) is an (ε, ε′ , K, DG′ )-framework where DG′ := D − 2r ∗ ≥ D − 2εn. Indeed, since (F, G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)weak framework, (FR1) and (FR3)–(FR5) follow from (WF1) and (WF3)–(WF5). Further, (FR2) follows from (WF2) and (c) while (FR6) follows from (b). (WF6) implies that all vertices v ∈ V (G) have internal degree at most dG (v)/2 in G. Thus all vertices v ∈ V (G′ ) have internal degree at most dG (v)/2 ≤ (dG′ (v) + 2r ∗ )/2 ≤ dG′ (v)/2 + εn in G′ . So (FR7) is satisfied. Hence, (i) is satisfied. Note that by definition of G′ , H contains all edges of G − G′ . So since r ∗ = (D − DG′ )/2 ≤ εn, (d) implies (ii). The following result follows immediately from Lemmas 4.4 and 4.12.

Corollary 4.13. Let 0 < 1/n ≪ ε ≪ ε∗ ≪ ε′ , 1/K ≪ α ≪ 1 and let D ≥ n/100. Suppose that F is an ε-bipartite graph on n vertices with δ(F ) ≥ (1/4+α)n. Suppose that G is a D-regular spanning subgraph of F . Then the following properties hold: (i) there is an (ε∗ , ε′ , K, DG′ )-framework (G′ , A, A0 , B, B0 ) such that G′ is a spanning subgraph of G, DG′ ≥ D − 2ε1/3 n and such that F satisfies (WF5) (with respect to the partition A, A0 , B, B0 ); (ii) there is a set of (D − DG′ )/2 ≤ ε1/3 n edge-disjoint Hamilton cycles in F − G′ containing all edges of G − G′ . In particular, if D is even then DG′ is even. 5. Finding path systems which cover all the edges within the classes The purpose of this section is to prove Corollary 5.11 which, given a framework (G, A, A0 , B, B0 ), guarantees a set C of edge-disjoint Hamilton cycles and a set J of suitable edge-disjoint 2-balanced A0 B0 -path systems such that the graph G∗ obtained from G by deleting the edges in all these Hamilton cycles and path systems is bipartite with vertex classes A′ and B ′ and A0 ∪ B0 is isolated in G∗ . Each of the path systems in J will later be extended into a Hamilton cycle by adding suitable edges between A and B. The path systems in J will need to be ‘localized’ with respect to a given partition. We prepare the ground for this in the next subsection. Throughout this section, given sets S, S ′ ⊆ V (G) we often write E(S), E(S, S ′ ), e(S) and e(S, S ′ ) for EG (S), EG (S, S ′ ), eG (S) and eG (S, S ′ ) respectively.


5.1. Choosing the partition and the localized slices. Let K, m ∈ N and ε > 0. A (K, m, ε)-partition of a set V of vertices is a partition of V into sets A0 , A1 , . . . , AK and B0 , B1 , . . . , BK such that |Ai | = |Bi | = m for all 1 ≤ i ≤ K and |A0 ∪ B0 | ≤ ε|V |. We often write V0 for A0 ∪ B0 and think of the vertices in V0 as ‘exceptional vertices’. The sets A1 , . . . , AK and B1 , . . . , BK are called clusters of the (K, m, ε0 )partition and A0 , B0 are called exceptional sets. Unless stated otherwise, when considering a (K, m, ε)-partition P we denote the elements of P by A0 , A1 , . . . , AK and B0 , B1 , . . . , BK as above. Further, we will often write A for A1 ∪ · · · ∪ AK and B for B1 ∪ · · · ∪ BK . Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and that ε1 , ε2 > 0. We say that P is a (K, m, ε, ε1 , ε2 )-partition for G if P satisfies the following properties: (P1) P is a (K, m, ε)-partition of V (G) such that the exceptional sets A0 and B0 in the partition P are the same as the sets A0 , B0 which are part of the framework (G, A, A0 , B, B0 ). In particular, m = |A|/K = |B|/K; (P2) d(v, Ai ) = (d(v, A) ± ε1 n)/K for all 1 ≤ i ≤ K and v ∈ V (G); (P3) e(Ai , Aj ) = 2(e(A) ± ε2 max{n, e(A)})/K 2 for all 1 ≤ i < j ≤ K; (P4) e(Ai ) = (e(A) ± ε2 max{n, e(A)})/K 2 for all 1 ≤ i ≤ K; (P5) e(A0 , Ai ) = (e(A0 , A) ± ε2 max{n, e(A0 , A)})/K for all 1 ≤ i ≤ K; (P6) e(Ai , Bj ) = (e(A, B) ± 3ε2 e(A, B))/K 2 for all 1 ≤ i, j ≤ K; and the analogous assertions hold if we replace A by B (as well as Ai by Bi etc.) in (P2)–(P5). Our first aim is to show that for every framework we can find such a partition with suitable parameters (see Lemma 5.2). To do this, we need the following lemma. Lemma 5.1. Suppose that 0 < 1/n ≪ ε, ε1 ≪ ε2 ≪ 1/K ≪ 1, that r ≤ 2K, that Km ≥ n/4 and that r, K, n, m ∈ N. Let G and F be graphs on n vertices with V (G) = V (F ). Suppose that there is a vertex partition of V (G) into U, R1 , . . . , Rr with the following properties: • |U | = Km. • δ(G[U ]) ≥ εn or ∆(G[U ]) ≤ εn. • For each j ≤ r we either have dG (u, Rj ) ≤ εn for all u ∈ U or dG (x, U ) ≥ εn for all x ∈ Rj . Then there exists a partition of U into K parts U1 , . . . , UK satisfying the following properties: (i) |Ui | = m for all i ≤ K. (ii) dG (v, Ui ) = (dG (v, U ) ± ε1 n)/K for all v ∈ V (G) and all i ≤ K. (iii) eG (Ui , Ui′ ) = 2(eG (U ) ± ε2 max{n, eG (U )})/K 2 for all 1 ≤ i 6= i′ ≤ K. (iv) eG (Ui ) = (eG (U ) ± ε2 max{n, eG (U )})/K 2 for all i ≤ K. (v) eG (Ui , Rj ) = (eG (U, Rj ) ± ε2 max{n, eG (U, Rj )})/K for all i ≤ K and j ≤ r. (vi) dF (v, Ui ) = (dF (v, U ) ± ε1 n)/K for all v ∈ V (F ) and all i ≤ K. Proof. Consider an equipartition U1 , . . . , UK of U which is chosen uniformly at random. So (i) holds by definition. Note that for a given vertex v ∈ V (G), dG (v, Ui ) has the hypergeometric distribution with mean dG (v, U )/K. So if dG (v, U ) ≥ ε1 n/K,


Proposition 2.3 implies that 2 ε1 dG (v, U ) 1 dG (v, U ) ε1 dG (v, U ) ≥ ≤ 2 exp − ≤ 2. P dG (v, Ui ) − K K 3K n Thus we deduce that for all v ∈ V (G) and all i ≤ K,

P (|dG (v, Ui ) − dG (v, U )/K| ≥ ε1 n/K) ≤ 1/n2 .

Similarly, P (|dF (v, Ui ) − dF (v, U )/K| ≥ ε1 n/K) ≤ 1/n2 .

So with probability at least 3/4, both (ii) and (vi) are satisfied. We now consider (iii) and (iv). Fix i, i′ ≤ K. If i 6= i′ , let X := eG (Ui , Ui′ ). If i = i′ , let X := 2eG (Ui ). For an edge f ∈ E(G[U ]), let Ef denote the event that f ∈ E(Ui , Ui′ ). So if f = xy and i 6= i′ , then m m (5.1) P(Ef ) = 2P(x ∈ Ui )P(y ∈ Ui′ | x ∈ Ui ) = 2 · . |U | |U | − 1 Similarly, if f and f ′ are disjoint (that is, f and f ′ have no common endpoint) and i 6= i′ , then (5.2)

P(Ef ′ | Ef ) = 2

m m m−1 m−1 · ≤2 · = P(Ef ′ ). |U | − 2 |U | − 3 |U | |U | − 1

By (5.1), if i 6= i′ , we also have (5.3)

E(X) = 2

eG (U ) |U | · = K2 |U | − 1

If f = xy and i = i′ , then

1±

2 |U |

2eG (U ) 2eG (U ) = (1 ± ε2 /4) . K2 K2

P(Ef ) = P(x ∈ Ui )P(y ∈ Ui | x ∈ Ui ) =

(5.4)

m m−1 · . |U | |U | − 1

So if i = i′ , similarly to (5.2) we also obtain P(Ef ′ | Ef ) ≤ P(Ef ) for disjoint f and f ′ and we obtain the same bound as in (5.3) on E(X) (recall that X = 2eG (Ui ) in this case). Note that if i 6= i′ then X X P(Ef ∩ Ef ′ ) − P(Ef )P(Ef ′ ) Var(X) = f ∈E(U ) f ′ ∈E(U )

=

X

P(Ef )

f ∈E(U ) (5.2)

≤

X

f ∈E(U )

X

f ′ ∈E(U )

P(Ef ′ | Ef ) − P(Ef ′ )

(5.3) 3e (U ) G · 2∆(G[U ]) ≤ eG (U )∆(G[U ]). P(Ef ) · 2∆(G[U ]) ≤ K2

Similarly, if i = i′ then X X Var(X) = 4

f ∈E(U ) f ′ ∈E(U )

P(Ef ∩ Ef ′ ) − P(Ef )P(Ef ′ ) ≤ eG (U )∆(G[U ]).


Let a := eG (U )∆(G[U ]). In both cases, from Chebyshev’s inequality, it follows that q P |X − E(X)| ≥ a/ε1/2 ≤ ε1/2 . p 1/2 ≤ Suppose that ∆(G[U ]) ≤ εn. If we also p have have eG (U ) ≤ n, then a/ε ε1/4 n ≤ ε2 n/2K 2 . If eG (U ) ≥ n, then a/ε1/2 ≤ ε1/4 eG (U ) ≤ ε2 eG (U )/2K 2 . If we do not have ∆(G[U ]) ≤ εn, then our assumptions imply that δ(G[U ]) ≥ εn. p So ∆(G[U ]) ≤ n ≤ εeG (G[U ]) with room to spare. This in turn means that a/ε1/2 ≤ ε1/4 eG (U ) ≤ ε2 eG (U )/2K 2 . So in all cases, we have ε2 max{n, eG (U )} (5.5) ≤ ε1/2 . P |X − E(X)| ≥ 2K 2 Now note that by (5.3) we have 2eG (U ) ε2 eG (U ) (5.6) E(X) − K 2 ≤ 2K 2 .

So (5.5) and (5.6) together imply that for fixed i, i′ the bound in (iii) fails with probability at most ε1/2 . The analogue holds for the bound in (iv). By summing over all possible values of i, i′ ≤ K, we have that (iii) and (iv) hold with probability at least 3/4. A similar argument shows that for all i ≤ K and j ≤ r, we have eG (U, Rj ) ε2 max{n, eG (U, Rj )} ≤ ε1/2 . (5.7) P eG (Ui , Rj ) − ≥ K K

Indeed, fix i ≤ K, j ≤ r and let X := eG (Ui , Rj ). For an edge f ∈ G[U, Rj ], let Ef denote the event that f ∈ E(Ui , Rj ). Then P(Ef ) = m/|U | = 1/K and so E(X) = eG (U, Rj )/K. The remainder of the argument proceeds as in the previous case (with slightly simpler calculations). So (v) holds with probability at least 3/4, by summing over all possible values of i ≤ K and j ≤ r again. So with positive probability, the partition satisfies all requirements. Lemma 5.2. Let 0 < 1/n ≪ ε ≪ ε′ ≪ ε1 ≪ ε2 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and δ(G) ≥ D ≥ n/200. Suppose that F is a graph with V (F ) = V (G). Then there exists a partition P = {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } of V (G) so that (i) P is a (K, m, ε, ε1 , ε2 )-partition for G. (ii) dF (v, Ai ) = (dF (v, A) ± ε1 n)/K and dF (v, Bi ) = (dF (v, B) ± ε1 n)/K for all 1 ≤ i ≤ K and v ∈ V (G). Proof. In order to find the required partitions A1 , . . . , AK of A and B1 , . . . , BK of B we will apply Lemma 5.1 twice, as follows. In the first application we let U := A, R1 := A0 , R2 := B0 and R3 := B. Note that ∆(G[U ]) ≤ ε′ n by (FR5) and dG (u, Rj ) ≤ |Rj | ≤ εn ≤ ε′ n for all u ∈ U and j = 1, 2 by (FR4). Moreover, (FR4) and (FR7) together imply that dG (x, U ) ≥ D/3 ≥ ε′ n for each x ∈ R3 = B. Thus we


can apply Lemma 5.1 with ε′ playing the role of ε to obtain a partition U1 , . . . , UK of U . We let Ai := Ui for all i ≤ K. Then the Ai satisfy (P2)–(P5) and (5.8)

eG (Ai , B) = (eG (A, B) ± ε2 max{n, eG (A, B)})/K = (1 ± ε2 )eG (A, B)/K.

Further, Lemma 5.1(vi) implies that

dF (v, Ai ) = (dF (v, A) ± ε1 n)/K

for all 1 ≤ i ≤ K and v ∈ V (G). For the second application of Lemma 5.1 we let U := B, R1 := B0 , R2 := A0 and Rj := Aj−2 for all 3 ≤ j ≤ K + 2. As before, ∆(G[U ]) ≤ ε′ n by (FR5) and dG (u, Rj ) ≤ εn ≤ ε′ n for all u ∈ U and j = 1, 2 by (FR4). Moreover, (FR4) and (FR7) together imply that dG (x, U ) ≥ D/3 ≥ ε′ n for all 3 ≤ j ≤ K + 2 and each x ∈ Rj = Aj−2 . Thus we can apply Lemma 5.1 with ε′ playing the role of ε to obtain a partition U1 , . . . , UK of U . Let Bi := Ui for all i ≤ K. Then the Bi satisfy (P2)–(P5) with A replaced by B, Ai replaced by Bi , and so on. Moreover, for all 1 ≤ i, j ≤ K, eG (Ai , Bj )

= (eG (Ai , B) ± ε2 max{n, eG (Ai , B)})/K (5.8) = ((1 ± ε2 )eG (A, B) ± ε2 (1 + ε2 )eG (A, B))/K 2 =

(eG (A, B) ± 3ε2 eG (A, B))/K 2 ,

i.e. (P6) holds. Since clearly (P1) holds as well, A0 , A1 , . . . , AK and B0 , B1 , . . . , BK together form a (K, m, ε, ε1 , ε2 )-partition for G. Further, Lemma 5.1(vi) implies that dF (v, Bi ) = (dF (v, B) ± ε1 n)/K

for all 1 ≤ i ≤ K and v ∈ V (G).

The next lemma gives a decomposition of G[A′ ] and G[B ′ ] into suitable smaller A and H B . We say that the graphs H A and H B guaranteed edge-disjoint subgraphs Hij ij ij ij by Lemma 5.3 are localized slices of G. Note that the order of the indices i and j A 6= H A . Also, we allow i = j. matters here, i.e. Hij ji Lemma 5.3. Let 0 < 1/n ≪ ε ≪ ε′ ≪ ε1 ≪ ε2 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and D ≥ n/200. Let A0 , A1 , . . . , AK and B0 , B1 , . . . , BK be a (K, m, ε, ε1 , ε2 )-partition for G. Then for A and H B with the following properties: all 1 ≤ i, j ≤ K there are graphs Hij ij

A is a spanning subgraph of G[A , A ∪ A ] ∪ G[A , A ] ∪ G[A ]; Hij 0 i j i j 0 A The sets E(Hij ) over all 1 ≤ i, j ≤ K form a partition of the edges of G[A′ ]; A ) = (e(A′ ) ± 9ε max{n, e(A′ )})/K 2 for all 1 ≤ i, j ≤ K; e(Hij 2 eH A (A0 , Ai ∪ Aj ) = (e(A0 , A) ± 2ε2 max{n, e(A0 , A)})/K 2 for all 1 ≤ i, j ≤ ij K; (v) eH A (Ai , Aj ) = (e(A) ± 2ε2 max{n, e(A)})/K 2 for all 1 ≤ i, j ≤ K; ij (vi) For all 1 ≤ i, j ≤ K and all v ∈ A0 we have dH A (v) = dH A (v, Ai ∪ Aj ) +

(i) (ii) (iii) (iv)

ij

dH A (v, A0 ) = (d(v, A) ± 4ε1 n)/K 2 .

ij

ij

The analogous assertions hold if we replace A by B, Ai by Bi , and so on.

´ ¨ 24 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN A we perform the following procedure: Proof. In order to construct the graphs Hij A is an empty graph with vertex set A ∪ A ∪ A . • Initially each Hij 0 i j • For all 1 ≤ i ≤ K choose a random partition E(A0 , Ai ) into K sets Uj of A ) := U . (If E(A , A ) is not divisible by K, first equal size and let E(Hij j 0 i distribute up to K − 1 edges arbitrarily among the Uj to achieve divisibility.) • For all i ≤ K, we add all the edges in E(Ai ) to HiiA . A • For all i, j ≤ K with i 6= j, half of the edges in E(Ai , Aj ) are added to Hij A and the other half is added to Hji (the choice of the edges is arbitrary). A . (So e • The edges in G[A0 ] are distributed equally amongst the Hij H A (A0 ) = ij

e(A0 )/K 2 ± 1.) Clearly, the above procedure ensures that properties (i) and (ii) hold. (P5) implies (iv) and (P3) and (P4) imply (v). Consider any v ∈ A0 . To prove (vi), note that we may assume that d(v, A) ≥ ε1 n/K 2 . Let X := dH A (v, Ai ∪ Aj ). Note that (P2) implies that E(X) = (d(v, A) ± ij

2ε1 n)/K 2 and note that E(X) ≤ n. So the Chernoff-Hoeffding bound for the hypergeometric distribution in Proposition 2.3 implies that 2

4

P(|X − E(X)| > ε1 n/K 2 ) ≤ P(|X − E(X)| > ε1 E(X)/K 2 ) ≤ 2e−ε1 E(X)/3K ≤ 1/n2 .

Since dH A (v, A0 ) ≤ |A0 | ≤ ε1 n/K 2 , a union bound implies the desired result. Finally, ij observe that for any a, b1 , . . . , b4 > 0, we have 4 X i=1

max{a, bi } ≤ 4 max{a, b1 , . . . , b4 } ≤ 4 max{a, b1 + · · · + b4 }.

So (iii) follows from (iv), (v) and the fact that eH A (A0 ) = e(A0 )/K 2 ± 1. ij

A will contain edges Note that the construction implies that if i 6= j, then Hij between A0 and Ai but not between A0 and Aj . However, this additional information is not needed in the subsequent argument.

5.2. Decomposing the localized slices. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework. Recall that a = |A0 |, b = |B0 | and a ≥ b. Since G is D-balanced by (FR2), we have e(A′ ) − e(B ′ ) = (a − b)D/2. So there are an integer q ≥ −b and a constant 0 ≤ c < 1 such that (5.9)

e(A′ ) = (a + q + c)D/2

and e(B ′ ) = (b + q + c)D/2.

The aim of this subsection is to prove Lemma 5.6, which guarantees a decomposition A into path systems (which will be extended into A B -path of each localized slice Hij 0 0 systems in Section 5.4) and a sparse (but not too sparse) leftover graph GA ij . The following two results will be used in the proof of Lemma 5.6. Lemma 5.4. Let 0 < 1/n ≪ α, β, γ so that γ < 1/2. Suppose that G is a graph on n vertices such that ∆(G) ≤ αn and e(G) ≥ βn. Then G contains a spanning subgraph H such that e(H) = ⌈(1 − γ)e(G)⌉ and ∆(G − H) ≤ 6γαn/5.


Proof. Let H ′ be a spanning subgraph of G such that • ∆(H ′ ) ≤ 6γαn/5; • e(H ′ ) ≥ γe(G).

To see that such a graph H ′ exists, consider a random subgraph of G obtained by including each edge of G with probability 11γ/10. Then E(∆(H ′ )) ≤ 11γαn/10 and E(e(H ′ )) = 11γe(G)/10. Thus applying Proposition 2.3 we have that, with high probability, H ′ is as desired. Define H to be a spanning subgraph of G such that H ⊇ G − H ′ and e(H) = ⌈(1 − γ)e(G)⌉. Then ∆(G − H) ≤ ∆(H ′ ) ≤ 6γαn/5, as required. Lemma 5.5. Suppose that G is a graph such that ∆(G) ≤ D − 2 where D ∈ N is even. Suppose A0 , A is a partition of V (G) such that dG (x) ≤ D/2 − 1 for all x ∈ A and ∆(G[A0 ]) ≤ D/2 − 1. Then G has a decomposition into D/2 edge-disjoint path systems P1 , . . . , PD/2 such that the following conditions hold: (i) For each i ≤ D/2, any internal vertex on a path in Pi lies in A0 ; (ii) |e(Pi ) − e(Pj )| ≤ 1 for all i, j ≤ D/2. Proof. Let G1 be a maximal spanning subgraph of G under the constraints that G[A0 ] ⊆ G1 and ∆(G1 ) ≤ D/2 − 1. Note that G[A0 ] ∪ G[A] ⊆ G1 . Set G2 := G − G1 . So G2 only contains A0 A-edges. Further, since ∆(G) ≤ D − 2, the maximality of G1 implies that ∆(G2 ) ≤ D/2 − 1. Define an auxiliary graph G′ , obtained from G1 as follows: write A0 = {a1 , . . . , am }. Add a new vertex set A′0 = {a′1 , . . . , a′m } to G1 . For each i ≤ m and x ∈ A, we add an edge between a′i and x if and only if ai x is an edge in G2 . Thus G′ [A0 ∪ A] is isomorphic to G1 and G′ [A′0 , A] is isomorphic to G2 . By construction and since dG (x) ≤ D/2−1 for all x ∈ A, we have that ∆(G′ ) ≤ D/2−1. Hence, Proposition 4.5 implies that E(G′ ) can be decomposed into D/2 edge-disjoint matchings M1 , . . . , MD/2 such that ||Mi | − |Mj || ≤ 1 for all i, j ≤ D/2. By identifying each vertex a′i ∈ A′0 with the corresponding vertex ai ∈ A0 , M1 , . . . , MD/2 correspond to edge-disjoint subgraphs P1 , . . . , PD/2 of G such that • P1 , . . . , PD/2 together cover all the edges in G; • |e(Pi ) − e(Pj )| ≤ 1 for all i, j ≤ D/2.

Note that dMi (x) ≤ 1 for each x ∈ V (G′ ). Thus dPi (x) ≤ 1 for each x ∈ A and dPi (x) ≤ 2 for each x ∈ A0 . This implies that any cycle in Pi must lie in G[A0 ]. However, Mi is a matching and G′ [A′0 ] ∪ G′ [A0 , A′0 ] contains no edges. Therefore, Pi contains no cycle, and so Pi is a path system such that any internal vertex on a path in Pi lies in A0 . Hence P1 , . . . , PD/2 satisfy (i) and (ii). Lemma 5.6. Let 0 < 1/n ≪ ε ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and D ≥ n/200. Let A0 , A1 , . . . , AK and B0 , B1 , . . . , BK be a (K, m, ε, ε1 , ε2 )-partition for G. Let A be a localized slice of G as guaranteed by Lemma 5.3. Define c and q as in (5.9). Hij


Suppose that t := (1 − 20ε4 )D/2K 2 ∈ N. If e(B ′ ) ≥ ε3 n, set t∗ to be the largest integer which is at most ct and is divisible by K 2 . Otherwise, set t∗ := 0. Define  if e(A′ ) < ε3 n;  0 a−b if e(A′ ) ≥ ε3 n but e(B ′ ) < ε3 n; ℓa :=  a + q + c otherwise and

ℓb :=

0 if e(B ′ ) < ε3 n; b + q + c otherwise.

A has a decomposition into t edge-disjoint path systems P , . . . , P and a Then Hij 1 t A spanning subgraph Gij with the following properties:

(i) (ii) (iii) (iv)

For each s ≤ t, any internal vertex on a path in Ps lies in A0 ; e(P1 ) = √ · · · = e(Pt∗ ) = ⌈ℓa ⌉ and e(Pt∗ +1 ) = · · · = e(Pt ) = ⌊ℓa ⌋; e(Ps ) ≤ εn for every s ≤ t; 2 ∆(GA ij ) ≤ 13ε4 D/K .

The analogous assertion (with ℓa replaced by ℓb and A0 replaced by B0 ) holds for B of G. Furthermore, ⌈ℓ ⌉ − ⌈ℓ ⌉ = ⌊ℓ ⌋ − ⌊ℓ ⌋ = a − b. each localized slice Hij a a b b Proof. Note that (5.9) and√(FR3) together imply that ℓa D/2 ≤ (a + q + c)D/2 = e(A′ ) ≤ εn2 and so ⌈ℓa ⌉ ≤ εn. Thus (iii) will follow from (ii). So it remains to prove (i), (ii) and (iv). We split the proof into three cases. Case 1. e(A′ ) < ε3 n (FR2) and (FR4) imply that e(A′ ) − e(B ′ ) = (a − b)D/2 ≥ 0. So e(B ′ ) ≤ e(A′ ) < A B B ε3 n. Thus ℓa = ℓb = 0. Set GA ij := Hij and Gij := Hij . Therefore, (iv) is satisfied A ) ≤ e(A′ ) < ε n ≤ 13ε D/K 2 . Further, (i) and (ii) are vacuous (i.e. we set as ∆(Hij 3 4 each Ps to be the empty graph on V (G)). Note that a = b since otherwise a > b and therefore (FR2) implies that e(A′ ) ≥ (a−b)D/2 ≥ D/2 > ε3 n, a contradiction. Hence, ⌈ℓa ⌉−⌈ℓb ⌉ = ⌊ℓa ⌋−⌊ℓb ⌋ = 0 = a−b.

Case 2. e(A′ ) ≥ ε3 n and e(B ′ ) < ε3 n B Since ℓb = 0 in this case, we set GB ij := Hij and each Ps to be the empty graph on B . Further, V (G). Then as in Case 1, (i), (ii) and (iv) are satisfied with respect to Hij clearly ⌈ℓa ⌉ − ⌈ℓb ⌉ = ⌊ℓa ⌋ − ⌊ℓb ⌋ = a − b. Note that a > b since otherwise a = b and thus e(A′ ) = e(B ′ ) by (FR2), a contradiction to the case assumptions. Since e(A′ ) − e(B ′ ) = (a − b)D/2 by (FR2), Lemma 5.3(iii) implies that A e(Hij ) ≥ (1 − 9ε2 )e(A′ )/K 2 − 9ε2 n/K 2 ≥ (1 − 9ε2 )(a − b)D/(2K 2 ) − 9ε2 n/K 2

(5.10)

≥ (1 − ε3 )(a − b)D/(2K 2 ) > (a − b)t.

Similarly, Lemma 5.3(iii) implies that (5.11)

A e(Hij ) ≤ (1 + ε4 )(a − b)D/(2K 2 ).

Therefore, (5.10) implies that there exists a constant γ > 0 such that A (1 − γ)e(Hij ) = (a − b)t.


Since (1 − 19ε4 )(1 − ε3 ) > (1 − 20ε4 ), (5.10) implies that γ > 19ε4 ≫ 1/n. Further, since (1 + ε4 )(1 − 21ε4 ) < (1 − 20ε4 ), (5.11) implies that γ < 21ε4 . Note that (FR5), (FR7) and Lemma 5.3(vi) imply that (5.12)

A ∆(Hij ) ≤ (D/2 + 5ε1 n)/K 2 .

A contains a spanning subgraph H such that e(H) = Thus Lemma 5.4 implies that Hij A ) = (a − b)t and (1 − γ)e(Hij A ∆(Hij − H) ≤ 6γ(D/2 + 5ε1 n)/(5K 2 ) ≤ 13ε4 D/K 2 , A where the last inequality follows since γ < 21ε4 and ε1 ≪ 1. Setting GA ij := Hij − H implies that (iv) is satisfied. Our next task is to decompose H into t edge-disjoint path systems so that (i) and (ii) are satisfied. Note that (5.12) implies that A ∆(H) ≤ ∆(Hij ) ≤ (D/2 + 5ε1 n)/K 2 < 2t − 2.

Further, (FR4) implies that ∆(H[A0 ]) ≤ |A0 | ≤ εn < t − 1 and (FR5) implies that dH (x) ≤ ε′ n < t − 1 for all x ∈ A. Since e(H) = (a − b)t, Lemma 5.5 implies that H has a decomposition into t edge-disjoint path systems P1 , . . . , Pt satisfying (i) and so that e(Ps ) = a − b = ℓa for all s ≤ t. In particular, (ii) is satisfied.

Case 3. e(A′ ), e(B ′ ) ≥ ε3 n By definition of ℓa and ℓb , we have that ⌈ℓa ⌉ − ⌈ℓb ⌉ = ⌊ℓa ⌋ − ⌊ℓb ⌋ = a − b. Notice that since e(A′ ) ≥ ε3 n and ε2 ≪ ε3 , certainly ε3 e(A′ )/(2K 2 ) > 9ε2 n/K 2 . Therefore, Lemma 5.3(iii) implies that

(5.13)

A e(Hij ) ≥ (1 − 9ε2 )e(A′ )/K 2 − 9ε2 n/K 2

≥ (1 − ε3 )e(A′ )/K 2 ≥ ε3 n/(2K 2 ).

Note that 1/n ≪ ε3 /(2K 2 ). Further, (5.9) and (5.13) imply that (5.14)

A e(Hij ) ≥ (1 − ε3 )e(A′ )/K 2

= (1 − ε3 )(a + q + c)D/(2K 2 ) > (a + q)t + t∗ .

Similarly, Lemma 5.3(iii) implies that (5.15)

A e(Hij ) ≤ (1 + ε3 )(a + q + c)D/(2K 2 ).

By (5.14) there exists a constant γ > 0 such that A (1 − γ)e(Hij ) = (a + q)t + t∗ .

Note that (5.14) implies that 1/n ≪ 19ε4 < γ and (5.15) implies that γ < 21ε4 . Moreover, as in Case 2, (FR5), (FR7) and Lemma 5.3(vi) together show that (5.16)

A ∆(Hij ) ≤ (D/2 + 5ε1 n)/K 2 .

´ ¨ 28 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN A contains a spanning subgraph Thus (as in Case 2 again), Lemma 5.4 implies that Hij A ) = (a + q)t + t∗ and H such that e(H) = (1 − γ)e(Hij A ∆(Hij − H) ≤ 6γ(D/2 + 5ε1 n)/(5K 2 ) ≤ 13ε4 D/K 2 . A Setting GA ij := Hij − H implies that (iv) is satisfied. Next we decompose H into t edge-disjoint path systems so that (i) and (ii) are satisfied. Note that (5.16) implies that A ∆(H) ≤ ∆(Hij ) ≤ (D/2 + 5ε1 n)/K 2 < 2t − 2.

Further, (FR4) implies that ∆(H[A0 ]) ≤ |A0 | ≤ εn < t − 1 and (FR5) implies that dH (x) ≤ ε′ n < t − 1 for all x ∈ A. Since e(H) = (a + q)t + t∗ , Lemma 5.5 implies that H has a decomposition into t edge-disjoint path systems P1 , . . . , Pt satisfying (i) and (ii). An identical argument implies that (i), (ii) and (iv) are satisfied with B also. respect to Hij

A 5.3. Decomposing the global graph. Let GA glob be the union of the graphs Gij B guaranteed by Lemma 5.6 over all 1 ≤ i, j ≤ K. Define Gglob similarly. The next B lemma gives a decomposition of both GA glob and Gglob into suitable path systems. Properties (iii) and (iv) of the lemma guarantee that one can pair up each such path B system QA ⊆ GA glob with a different path system QB ⊆ Gglob such that QA ∪ QB is 2-balanced (in particular e(QA ) − e(QB ) = a − b). This property will then enable us to apply Lemma 4.10 to extend QA ∪ QB into a Hamilton cycle using only edges between A′ and B ′ .

Lemma 5.7. Let 0 < 1/n ≪ ε ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and such that D ≥ n/200 and D is even. Let A0 , A1 , . . . , AK and B0 , B1 , . . . , BK be a (K, m, ε, ε1 , ε2 )A partition for G. Let GA glob be the union of the graphs Gij guaranteed by Lemma 5.6 over all 1 ≤ i, j ≤ K. Define GB glob similarly. Suppose that k := 10ε4 D ∈ N. Then the following properties hold: (i) There is an integer q ′ and a real number 0 ≤ c′ < 1 so that e(GA glob ) = ′ ′ ′ ′ B (a + q + c )k and e(Gglob ) = (b + q + c )k. B (ii) ∆(GA glob ), ∆(Gglob ) < 3k/2. ∗ (iii) Let k∗ := c′ k. Then GA glob has a decomposition into k path systems, each ′ ∗ containing a + q + 1 edges, and k − k path systems, each containing a + q ′ edges. Moreover, each of these k path systems Q satisfies dQ (x) ≤ 1 for all x ∈ A. ∗ ′ (iv) GB glob has a decomposition into k path systems, each containing b + q + 1 edges, and k − k∗ path systems, each containing b + q ′ edges. Moreover, each of these k path systems Q satisfies dQ (x) ≤ 1 for all x ∈ B. √ (v) Each of the path systems guaranteed in (iii) and (iv) contains at most εn edges.


Note that in Lemma 5.7 and several later statements the parameter ε3 is implicitly B defined by the application of Lemma 5.6 which constructs the graphs GA glob and Gglob . Proof. Let t∗ and t be as defined in Lemma 5.6. Our first task is to show that (i) ′ ′ B is satisfied. If e(A′ ), e(B ′ ) < ε3 n then GA glob = G[A ] and Gglob = G[B ]. Further, a = b in this case since otherwise (FR4) implies that a > b and so (FR2) yields that e(A′ ) ≥ (a − b)D/2 ≥ D/2 > ε3 n, a contradiction. Therefore, (FR2) implies that B ′ ′ e(GA glob ) − e(Gglob ) = e(A ) − e(B )=(a − b)D/2 = 0 = (a − b)k.

′ A If e(A′ ) ≥ ε3 n and e(B ′ ) < ε3 n then GB glob = G[B ]. Further, Gglob is obtained from G[A′ ] by removing tK 2 edge-disjoint path systems, each of which contains precisely a − b edges. Thus (FR2) implies that ′ ′ 2 2 B e(GA glob ) − e(Gglob ) = e(A ) − e(B ) − tK (a − b) = (a − b)(D/2 − tK ) = (a − b)k.

Finally, consider the case when e(A′ ), e(B ′ ) > ε3 n. Then GA glob is obtained from ′ ∗ 2 G[A ] by removing t K edge-disjoint path systems, each of which contain exactly a + q + 1 edges, and by removing (t − t∗ )K 2 edge-disjoint path systems, each of which ′ ∗ 2 contain exactly a+q edges. Similarly, GB glob is obtained from G[B ] by removing t K edge-disjoint path systems, each of which contain exactly b + q + 1 edges, and by removing (t − t∗ )K 2 edge-disjoint path systems, each of which contain exactly b + q edges. So (FR2) implies that B ′ ′ 2 e(GA glob ) − e(Gglob ) = e(A ) − e(B ) − (a − b)tK = (a − b)k.

Therefore, in every case, (5.17)

B e(GA glob ) − e(Gglob ) = (a − b)k.

′ ′ Define the integer q ′ and 0 ≤ c′ < 1 by e(GA glob ) = (a + q + c )k. Then (5.17) implies ′ ′ that e(GB glob ) = (b + q + c )k. This proves (i). To prove (ii), note that Lemma 5.6(iv) B implies that ∆(GA glob ) ≤ 13ε4 D < 3k/2 and similarly ∆(Gglob ) < 3k/2. ′ Note that (FR5) implies that dGA (x) ≤ ε n < k−1 for all x ∈ A and ∆(GA glob [A0 ]) ≤ glob |A0 | ≤ εn < k − 1. Thus Lemma 5.5 together with (i) implies that (iii) is satisfied. (iv) follows from Lemma 5.5 analogously. ′ 2 . There′ 2 B (FR3) implies that e(GA glob ) ≤ eG (A ) ≤ εn and e(Gglob ) ≤ eG (B ) ≤ εn √ fore, each path system from (iii) and (iv) contains at most ⌈εn2 /k⌉ ≤ εn edges. So (v) is satisfied.

We say that a path system P ⊆ G[A′ ] is (i, j, A)-localized if (i) E(P ) ⊆ E(G[A0 , Ai ∪ Aj ]) ∪ E(G[Ai , Aj ]) ∪ E(G[A0 ]); (ii) Any internal vertex on a path in P lies in A0 . We introduce an analogous notion of (i, j, B)-localized for path systems P ⊆ G[B ′ ]. The following result is a straightforward consequence of Lemmas 5.3, 5.6 and 5.7. It gives a decomposition of G[A′ ] ∪ G[B ′ ] into pairs of paths systems so that most of these are localized and so that each pair can be extended into a Hamilton cycle by adding A′ B ′ -edges.


Corollary 5.8. Let 0 < 1/n ≪ ε ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and such that D ≥ n/200 and D is even. Let A0 , A1 , . . . , AK and B0 , B1 , . . . , BK be a (K, m, ε, ε1 , ε2 )partition for G. Let tK := (1 − 20ε4 )D/2K 4 and k := 10ε4 D. Suppose that tK ∈ N. Then there are K 4 sets Mi1 i2 i3 i4 , one for each 1 ≤ i1 , i2 , i3 , i4 ≤ K, such that each Mi1 i2 i3 i4 consists of tK pairs of path systems and satisfies the following properties: (a) Let (P, P ′ ) be a pair of path systems which forms an element of Mi1 i2 i3 i4 . Then (i) P is an (i1 , i2 , A)-localized path system and P ′ is an (i3 , i4 , B)-localized path system; (ii) e(P ) − e(P ′ ) = √a − b; ′ (iii) e(P ), e(P ) ≤ εn. (b) The 2tK path systems in the pairs belonging to Mi1 i2 i3 i4 are all pairwise edge-disjoint. (c) Let G(Mi1 i2 i3 i4 ) denote the spanning subgraph of G whose edge set is the union of all the path systems in the pairs belonging to Mi1 i2 i3 i4 . Then the K 4 graphs G(Mi1 i2 i3 i4 ) are edge-disjoint. Further, each x ∈ A0 satisfies dG(Mi1 i2 i3 i4 ) (x) ≥ (dG (x, A) − 15ε4 D)/K 4 while each y ∈ B0 satisfies dG(Mi1 i2 i3 i4 ) (y) ≥ (dG (y, B) − 15ε4 D)/K 4 . (d) Let Gglob be the subgraph of G[A′ ] ∪ G[B ′ ] obtained by removing all edges contained in G(Mi1 i2 i3 i4 ) for all 1 ≤ i1 , i2 , i3 , i4 ≤ K. Then ∆(Gglob ) ≤ 3k/2. Moreover, Gglob has a decomposition into k pairs of path systems (Q1,A , Q1,B ), . . . , (Qk,A , Qk,B ) so that (i′ ) Qi,A ⊆ Gglob [A′ ] and Qi,B ⊆ Gglob [B ′ ] for all i ≤ k; (ii′ ) dQi,A (x) ≤ 1 for all x ∈ A and dQi,B (x) ≤ 1 for all x ∈ B; (iii′ ) e(Qi,A ) − e(Qi,B ) = √a − b for all i ≤ k; (iv′ ) e(Qi,A ), e(Qi,B ) ≤ εn for all i ≤ k. A and H B (for all i, j ≤ K). Proof. Apply Lemma 5.3 to obtain localized slices Hij ij 2 ∗ Let t := K tK and let t be as defined in Lemma 5.6. Since t/K 2 , t∗ /K 2 ∈ N we have (t − t∗ )/K 2 ∈ N. For all i1 , i2 ≤ K, let MA i1 i2 be the set of t path systems in HiA1 i2 guaranteed by Lemma 5.6. We call the t∗ path systems in MA i1 i2 of size ⌈ℓa ⌉ B large and the others small. We define Mi3 i4 as well as large and small path systems in MB i3 i4 analogously (for all i3 , i4 ≤ K). We now construct the sets Mi1 i2 i3 i4 as follows: For all i1 , i2 ≤ K, consider a 2 random partition of the set of all large path systems in MA i1 i2 into K sets of equal size t∗ /K 2 and assign (all the path systems in) each of these sets to one of the Mi1 i2 i3 i4 with i3 , i4 ≤ K. Similarly, randomly partition the set of small path systems 2 ∗ 2 in MA i1 i2 into K sets, each containing (t−t )/K path systems. Assign each of these K 2 sets to one of the Mi1 i2 i3 i4 with i3 , i4 ≤ K. Proceed similarly for each MB i3 i4 in order to assign each of its path systems randomly to some Mi1 i2 i3 i4 . Then to each Mi1 i2 i3 i4 we have assigned exactly t∗ /K 2 large path systems from both MA i1 i2 and B Mi3 i4 . Pair these off arbitrarily. Similarly, pair off the small path systems assigned


to Mi1 i2 i3 i4 arbitrarily. Clearly, the sets Mi1 i2 i3 i4 obtained in this way satisfy (a) and (b). We now verify (c). By construction, the K 4 graphs G(Mi1 i2 i3 i4 ) are edge-disjoint. So consider any vertex x ∈ A0 and write d := dG (x, A). Note that dH A (x) ≥ i1 i2

(d − 4ε1 n)/K 2 by Lemma 5.3(vi). Let G(MA i1 i2 ) be the spanning subgraph of G whose edge set is the union of all the path systems in MA i1 i2 . Then Lemma 5.6(iv) implies that d − 14ε4 D d − 4ε1 n 13ε4 D − ≥ . dG(MA ) (x) ≥ dH A (x) − ∆(GA i1 i2 ) ≥ 2 2 i1 i2 i1 i2 K K K2 So a Chernoff-Hoeffding estimate for the hypergeometric distribution (Proposition 2.3) implies that d − 14ε4 D 1 d − 15ε4 D dG(Mi1 i2 i3 i4 ) (x) ≥ 2 . − εn ≥ 2 K K K4 (Note that we only need to apply the Chernoff-Hoeffding bound if d ≥ εn say, as (c) is vacuous otherwise.) It remains to check condition (d). First note that k ∈ N since tK , D/2 ∈ N. B Thus we can apply Lemma 5.7 to obtain a decomposition of both GA glob and Gglob B into path systems. Since Gglob = GA glob ∪ Gglob , (d) is an immediate consequence of Lemma 5.7(ii)–(v). 5.4. Constructing the localized balanced exceptional systems. The localized path systems obtained from Corollary 5.8 do not yet cover all of the exceptional vertices. This is achieved via the following lemma: we extend the path systems to achieve this additional property, while maintaining the property of being balanced. More precisely, let P := {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK }

be a (K, m, ε)-partition of a set V of n vertices. Given 1 ≤ i1 , i2 , i3 , i4 ≤ K and ε0 > 0, an (i1 , i2 , i3 , i4 )-balanced exceptional system with respect to P and parameter ε0 is a path system J with V (J) ⊆ A0 ∪ B0 ∪ Ai1 ∪ Ai2 ∪ Bi3 ∪ Bi4 such that the following conditions hold: (BES1) Every vertex in A0 ∪ B0 is an internal vertex of a path in J. Every vertex v ∈ Ai1 ∪ Ai2 ∪ Bi3 ∪ Bi4 satisfies dJ (v) ≤ 1. (BES2) Every edge of J[A ∪ B] is either an Ai1 Ai2 -edge or a Bi3 Bi4 -edge. (BES3) The edges in J cover precisely the same number of vertices in A as in B. (BES4) e(J) ≤ ε0 n. To shorten the notation, we will often refer to J as an (i1 , i2 , i3 , i4 )-BES. If V is the vertex set of a graph G and J ⊆ G, we also say that J is an (i1 , i2 , i3 , i4 )-BES in G. Note that (BES2) implies that an (i1 , i2 , i3 , i4 )-BES does not contain edges between A and B. Furthermore, an (i1 , i2 , i3 , i4 )-BES is also, for example, an (i2 , i1 , i4 , i3 )BES. We will sometimes omit the indices i1 , i2 , i3 , i4 and just refer to a balanced exceptional system (or a BES for short). We will sometimes also omit the partition P, if it is clear from the context.


(BES1) implies that each balanced exceptional system is an A0 B0 -path system as defined before Proposition 4.6. (However, the converse is not true since, for example, a 2-balanced A0 B0 -path system need not satisfy (BES4).) So (BES3) and Proposition 4.6 imply that each balanced exceptional system is also 2-balanced. We now extend each set Mi1 i2 i3 i4 obtained from Corollary 5.8 into a set Ji1 i2 i3 i4 of (i1 , i2 , i3 , i4 )-BES. Lemma 5.9. Let 0 < 1/n ≪ ε ≪ ε0 ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and such that D ≥ n/200 and D is even. Let P := {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } be a (K, m, ε, ε1 , ε2 )-partition for G. Suppose that tK := (1 − 20ε4 )D/2K 4 ∈ N. Let Mi1 i2 i3 i4 be the sets returned by Corollary 5.8. Then for all 1 ≤ i1 , i2 , i3 , i4 ≤ K there is a set Ji1 i2 i3 i4 which satisfies the following properties:

(i) Ji1 i2 i3 i4 consists of tK edge-disjoint (i1 , i2 , i3 , i4 )-BES in G with respect to P and with parameter ε0 . (ii) For each of the tK pairs of path systems (P, P ′ ) ∈ Mi1 i2 i3 i4 , there is a unique J ∈ Ji1 i2 i3 i4 which contains all the edges in P ∪ P ′ . Moreover, all edges in E(J) \ E(P ∪ P ′ ) lie in G[A0 , Bi3 ] ∪ G[B0 , Ai1 ]. (iii) Whenever (i1 , i2 , i3 , i4 ) 6= (i′1 , i′2 , i′3 , i′4 ), J ∈ Ji1 i2 i3 i4 and J ′ ∈ Ji′1 i′2 i′3 i′4 , then J and J ′ are edge-disjoint.

We let J denote the union of the sets Ji1 i2 i3 i4 over all 1 ≤ i1 , i2 , i3 , i4 ≤ K. Proof. We will construct the sets Ji1 i2 i3 i4 greedily by extending each pair of path systems (P, P ′ ) ∈ Mi1 i2 i3 i4 in turn into an (i1 , i2 , i3 , i4 )-BES containing P ∪ P ′ . For this, consider some arbitrary ordering of the K 4 4-tuples (i1 , i2 , i3 , i4 ). Suppose that we have already constructed the sets Ji′1 i′2 i′3 i′4 for all (i′1 , i′2 , i′3 , i′4 ) preceding (i1 , i2 , i3 , i4 ) so that (i)–(iii) are satisfied. So our aim now is to construct Ji1 i2 i3 i4 . Consider an enumeration (P1 , P1′ ), . . . , (PtK , Pt′K ) of the pairs of path systems in Mi1 i2 i3 i4 . Suppose that for some i ≤ tK we have already constructed edge-disjoint (i1 , i2 , i3 , i4 )-BES J1 , . . . , Ji−1 , so that for each i′ < i the following conditions hold: • Ji′ contains the edges in Pi′ ∪ Pi′′ ; • all edges in E(Ji′ ) \ E(Pi′ ∪ Pi′′ ) lie in G[A0 , Bi3 ] ∪ G[B0 , Ai1 ];S • Ji′ is edge-disjoint from all the balanced exceptional systems in (i′ ,i′ ,i′ ,i′ ) Ji′1 i′2 i′3 i′4 , 1 2 3 4 where the union is over all (i′1 , i′2 , i′3 , i′4 ) preceding (i1 , i2 , i3 , i4 ). We will now construct J := Ji . For this, we need to add suitable edges to Pi ∪ Pi′ to ensure that all vertices of A0 ∪ B0 have degree two. We start with A0 . Recall that a = |A0 | and write A0 = {x1 , . . . , xa }. Let G′ denote the subgraph of G[A′ , B ′ ] obtained by removing all the edges lying in J1 , . . . , Ji−1 S as well as all those edges lying in the balanced exceptional systems belonging to (i′ ,i′ ,i′ ,i′ ) Ji′1 i′2 i′3 i′4 (where as 1 2 3 4 before the union is over all (i′1 , i′2 , i′3 , i′4 ) preceding (i1 , i2 , i3 , i4 )). We will choose the new edges incident to A0 in J inside G′ [A0 , Bi3 ]. Suppose we have already found suitable edges for x1 , . . . , xj−1 and let J(j) be the set of all these edges. We will first show that the degree of xj inside G′ [A0 , Bi3 ] is still large. Let dj := dG (xj , A′ ). Consider any (i′1 , i′2 , i′3 , i′4 ) preceding (i1 , i2 , i3 , i4 ).


Let G(Ji′1 i′2 i′3 i′4 ) denote the union of the tK balanced exceptional systems belonging to Ji′1 i′2 i′3 i′4 . Thus dG(Ji′ i′ i′ i′ ) (xj ) = 2tK . However, Corollary 5.8(c) implies that 1 2 3 4

dG(Mi′ i′ i′ i′ ) (xj ) ≥ (dj −15ε4 D)/K 4 . So altogether, when constructing (the balanced 1 2 3 4

exceptional systems in) Ji′1 i′2 i′3 i′4 , we have added at most 2tK − (dj − 15ε4 D)/K 4 new edges at xj , and all these edges join xj to vertices in Bi′3 . Similarly, when constructing J1 , . . . , Ji−1 , we have added at most 2tK − (dj − 15ε4 D)/K 4 new edges at xj . Since the number of 4-tuples (i′1 , i′2 , i′3 , i′4 ) with i′3 = i3 is K 3 , it follows that dj − 15ε4 D 3 ′ dG (xj , Bi3 ) − dG (xj , Bi3 ) ≤ K 2tK − K4 1 = ((1 − 20ε4 )D − dj + 15ε4 D) K 1 (D − dj − 5ε4 D) . = K Also, (P2) with A replaced by B implies that dG (xj , Bi3 ) ≥

dG (xj , B) − ε1 n dG (xj ) − dG (xj , A′ ) − ε1 n D − dj − ε1 n ≥ = , K K K

where here we use (FR2) and (FR6). So altogether, we have dG′ (xj , Bi3 ) ≥ (5ε4 D − ε1 n)/K ≥ ε4 n/50K. Let Bi′3 be the set of vertices in Bi3 not covered by the edges of J(j) ∪ Pi′ . Note √ that |Bi′3 | ≥ |Bi3 | − 2|A0 | − 2e(Pi′ ) ≥ |Bi3 | − 3 εn since a = |A0 | ≤ εn by (FR4) √ and e(Pi′ ) ≤ εn by Corollary 5.8(a)(iii). So dG′ (xj , Bi′3 ) ≥ ε4 n/51K. We can add up to two of these edges to J in order to ensure that xj has degree two in J. This completes the construction of the edges of J incident to A0 . The edges incident to B0 are found similarly. Let J be the graph on A0 ∪ B0 ∪ Ai1 ∪ Ai2 ∪ Bi3 ∪ Bi4 whose edge set is constructed in this way. By construction, J satisfies (BES1) and (BES2) since Pj and Pj′ are (i1 , i2 , A)-localized and (i3 , i4 , B)-localized respectively. We now verify (BES3). As mentioned before the statement of the lemma, (BES1) implies that J is an A0 B0 -path system (as defined before Proposition 4.6). Moreover, Corollary 5.8(a)(ii) implies that Pi ∪ Pi′ is a path system which satisfies (B1) in the definition of 2-balanced. Since J was obtained by adding only A′ B ′ -edges, (B1) is preserved in J. Since by construction J satisfies (B2), it follows that J is 2-balanced. So Proposition 4.6 implies (BES3). Finally, we verify (BES4). For this, note that Corollary 5.8(a)(iii) implies that √ e(Pi ), e(Pi′ ) ≤ εn. Moreover, the number of edges added to Pi ∪ Pi′ when constructing J is at most 2(|A0 | + |B0 |), which is at most 2εn by (FR4). Thus e(J) ≤ √ 2 εn + 2εn ≤ ε0 n.


5.5. Covering Gglob by edge-disjoint Hamilton cycles. We now find a set of edge-disjoint Hamilton cycles covering the edges of the ‘leftover’ graph obtained from G − G[A, B] by deleting all those edges lying in balanced exceptional systems belonging to J. Lemma 5.10. Let 0 < 1/n ≪ ε ≪ ε0 ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and such that D ≥ n/200 and D is even. Let P := {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } be a (K, m, ε, ε1 , ε2 )-partition for G. Suppose that tK := (1 − 20ε4 )D/2K 4 ∈ N. Let J be as defined after Lemma 5.9 and let G(J) ⊆ G be the union of all the balanced exceptional systems lying in J. Let G∗ := G − G(J), let k := 10ε4 D and let (Q1,A , Q1,B ), . . . , (Qk,A , Qk,B ) be as in Corollary 5.8(d). (a) The graph G∗ −G∗ [A, B] can be decomposed into k A0 B0 -path systems Q1 , . . . , Qk which are 2-balanced and satisfy the following properties: (i) Qi contains all edges of Qi,A ∪ Qi,B ; (ii) Q1 , . . . , Q√ k are pairwise edge-disjoint; (iii) e(Qi ) ≤ 3 εn. (b) Let Q1 , . . . , Qk be as in (a). Suppose that F is a graph on V (G) such that G ⊆ F , δ(F ) ≥ 2n/5 and such that F satisfies (WF5) with respect to ε′ . Then there are edge-disjoint Hamilton cycles C1 , . . . , Ck in F − G(J) such that Qi ⊆ Ci and Ci ∩ G is 2-balanced for each i ≤ k. Proof. We first prove (a). The argument is similar to that of Lemma 5.7. Roughly speaking, we will extend each Qi,A into a path system Q′i,A by adding suitable A0 Bedges which ensure that every vertex in A0 has degree exactly two in Q′i,A . Similarly, we will extend each Qi,B into Q′i,B by adding suitable AB0 -edges. We will ensure that no vertex is an endvertex of both an edge in Q′i,A and an edge in Q′i,B and take Qi to be the union of these two path systems. We first construct all the Q′i,A . Claim 1. G∗ [A′ ] ∪ G∗ [A0 , B] has a decomposition into edge-disjoint path systems Q′1,A , . . . , Q′k,A such that • Qi,A ⊆ Q′i,A and E(Q′i,A ) \ E(Qi,A ) consists of A0 B-edges in G∗ (for each i ≤ k); / A0 ; • dQ′i,A (x) = 2 for every x ∈ A0 and dQ′i,A (x) ≤ 1 for every x ∈ ′ • no vertex is an endvertex of both an edge in Qi,A and an edge in Qi,B (for each i ≤ k). To prove Claim 1, let Gglob be as defined in Corollary 5.8(d). Thus Gglob [A′ ] = Q1,A ∪ · · · ∪ Qk,A. On the other hand, Lemma 5.9(ii) implies that G∗ [A′ ] = Gglob [A′ ]. Hence, (5.18)

G∗ [A′ ] = Gglob [A′ ] = Q1,A ∪ · · · ∪ Qk,A .

Similarly, G∗ [B ′ ] = Gglob [B ′ ] = Q1,B ∪ · · · ∪ Qk,B . Moreover, Gglob = G∗ [A′ ] ∪ G∗ [B ′ ]. Consider any vertex x ∈ A0 . Let dglob (x) denote the degree of x in Q1,A ∪ · · · ∪ Qk,A.


So dglob (x) = dG∗ (x, A′ ) by (5.18). Let dloc (x) := dG (x, A′ ) − dglob (x)

(5.19)

= dG (x, A′ ) − dG∗ (x, A′ ) = dG(J) (x, A′ ).

(5.20) Then

dloc (x) + dG (x, B ′ ) + dglob (x)

(5.21)

(5.19)

=

dG (x) = D,

where the final equality follows from (FR2). Recall that J consists of K 4 tK edgedisjoint balanced exceptional systems. Since x has two neighbours in each of these balanced exceptional systems, the degree of x in G(J) is 2K 4 tK = D−2k. Altogether this implies that dG∗ (x, B ′ ) (5.22)

= (5.20)

=

dG (x, B ′ ) − dG(J) (x, B ′ ) = dG (x, B ′ ) − (dG(J) (x) − dG(J) (x, A′ )) dG (x, B ′ ) − (D − 2k − dloc (x))

(5.21)

=

2k − dglob (x).

Note that this is precisely the total number of edges at x which we need to add to Q1,A , . . . , Qk,A in order to obtain Q′1,A , . . . , Q′k,A as in Claim 1. We can now construct the path systems Q′i,A . For each x ∈ A0 , let ni (x) = 2 − dQi,A (x). So 0 ≤ ni (x) ≤ 2 for all i ≤ k. Recall that a := |A0 | and consider an ordering x1 , . . . , xa of the vertices in A0 . Let G∗j := G∗ [{x1 , . . . , xj }, B]. Assume that for some 0 ≤ j < a, we have already found a decomposition of G∗j into edge-disjoint path systems Q1,j , . . . , Qk,j satisfying the following properties (for all i ≤ k): (i′ ) no vertex is an endvertex of both an edge in Qi,j and an edge in Qi,B ; (ii′ ) xj ′ has degree ni (xj ′ ) in Qi,j for all j ′ ≤ j and all other vertices have degree at most one in Qi,j . We call this assertion Aj . We will show that Aj+1 holds (i.e. the above assertion also holds with j replaced by j + 1). This in turn implies Claim 1 if we let Q′i,A := Qi,a ∪ Qi,A for all i ≤ k. To prove Aj+1 , consider the following bipartite auxiliary graph Hj+1 . The vertex classes of Hj+1 are Nj+1 := NG∗ (xj+1 )∩ B and Zj+1 , where Zj+1 is a multiset whose elements are chosen from Q1,B , . . . , Qk,B . Each Qi,B is included exactly ni (xj+1 ) times in Zj+1 . Note that Nj+1 = NG∗ (xj+1 ) ∩ B ′ since e(G[A0 , B0 ]) = 0 by (FR6). Altogether this implies that (5.23) |Zj+1 |

=

k X i=1

(5.22)

=

ni (xj+1 ) = 2k −

k X i=1

dQi,A (xj+1 ) = 2k − dglob (xj+1 )

dG∗ (xj+1 , B ′ ) = |Nj+1 | ≥ k/2.

The final inequality follows from (5.22) since dglob (xj+1 )

(5.18)

≤

∆(Gglob [A′ ]) ≤ 3k/2

by Corollary 5.8(d). We include an edge in Hj+1 between v ∈ Nj+1 and Qi,B ∈ Zj+1 if v is not an endvertex of an edge in Qi,B ∪ Qi,j .

´ ¨ 36 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN ′ . Claim 2. Hj+1 has a perfect matching Mj+1

Given the perfect matching guaranteed by the claim, we construct Qi,j+1 from Qi,j as follows: the edges of Qi,j+1 incident to xj+1 are precisely the edges xj+1 v where vQi,B ′ is an edge of Mj+1 (note that there are up to two of these). Thus Claim 2 implies that Aj+1 holds. (Indeed, (i′ )–(ii′ ) are immediate from the definition of Hj+1 .)

To prove Claim 2, consider any vertex v ∈ Nj+1 . Since v ∈ B, the number of path systems Qi,B containing an edge at v is at most dG (v, B ′ ). The number of indices i for which Qi,j contains an edge at v is at most dG (v, A0 ) ≤ |A0 |. Since each path system Qi,B occurs at most twice in the multiset Zj+1 , it follows that the degree of v in Hj+1 is at least |Zj+1 | − 2dG (v, B ′ ) − 2|A0 |. Moreover, dG (v, B ′ ) ≤ ε′ n ≤ k/16 (say) by (FR5). Also, |A0 | ≤ εn ≤ k/16 by (FR4). So v has degree at least |Zj+1 | − k/4 ≥ |Zj+1 |/2 in Hj+1 . √ Now consider any path system Qi,B ∈ Zj+1 . Recall that e(Qi,B ) ≤ εn ≤ k/16 (say), where the first inequality follows from Corollary 5.8(d)(iv′ ). Moreover, e(Qi,j ) ≤ 2|A0 | ≤ 2εn ≤ k/8, where the second inequality follows from (FR4). Thus the degree of Qi,B in Hj+1 is at least |Nj+1 | − 2e(Qi,B ) − e(Qi,j ) ≥ |Nj+1 | − k/4 ≥ |Nj+1 |/2.

′ , as required. Altogether this implies that Hj+1 has a perfect matching Mj+1

This completes the construction of Q′1,A , . . . , Q′k,A . Next we construct Q′1,B , . . . , Q′k,B using the same approach. Claim 3. G∗ [B ′ ] ∪ G∗ [B0 , A] has a decomposition into edge-disjoint path systems Q′1,B , . . . , Q′k,B such that • Qi,B ⊆ Q′i,B and E(Q′i,B ) \ E(Qi,B ) consists of B0 A-edges in G∗ (for each i ≤ k); • dQ′i,B (x) = 2 for every x ∈ B0 and dQ′i,B (x) ≤ 1 for every x ∈ / B0 ; ′ • no vertex is an endvertex of both an edge in Qi,A and an edge in Q′i,B (for each i ≤ k). The proof of Claim 3 is similar to that of Claim 1. The only difference is that when constructing Q′i,B , we need to avoid the endvertices of all the edges in Q′i,A (not just the edges in Qi,A ). However, e(Q′i,A − Qi,A ) ≤ 2|A0 |, so this does not affect the calculations significantly. We now take Qi := Q′i,A ∪ Q′i,B for all i ≤ k. Then the Qi are pairwise edge-disjoint and √ √ e(Qi ) ≤ e(Qi,A ) + e(Qi,B ) + 2|A0 ∪ B0 | ≤ 2 εn + 2εn ≤ 3 εn by Corollary 5.8(d)(iv′ ) and (FR4). Moreover, Corollary 5.8(d)(iii′ ) implies that (5.24)

eQi (A′ ) − eQi (B ′ ) = e(Qi,A ) − e(Qi,B ) = a − b.

Thus each Qi is a 2-balanced A0 B0 -path system. Further, Q1 , . . . , Qk form a decomposition of G∗ [A′ ] ∪ G∗ [A0 , B] ∪ G∗ [B ′ ] ∪ G∗ [B0 , A] = G∗ − G∗ [A, B].


(The last equality follows since e(G[A0 , B0 ]) = 0 by (FR6).) This completes the proof of (a). To prove (b), note that (F, G, A, A0 , B, B0 ) is an (ε, ε′ , D)-pre-framework, i.e. it satisfies (WF1)–(WF5). Indeed, recall that (FR1)–(FR4) imply (WF1)–(WF4) and that (WF5) holds by assumption. So we can apply Lemma 4.10 (with Q1 playing the role of Q) to extend Q1 into a Hamilton cycle C1 . Moreover, Lemma 4.10(iii) implies that C1 ∩ G is 2-balanced, as required. (Lemma 4.10(ii) guarantees that C1 is edge-disjoint from Q2 , . . . , Qk and G(J).) Let G1 := G − C1 and F1 := F − C1 . Proposition 4.3 (with C1 playing the role of H) implies that (F1 , G1 , A, A0 , B, B0 ) is an (ε, ε′ , D − 2)-pre-framework. So we can now apply Lemma 4.10 to (F1 , G1 , A, A0 , B, B0 ) to extend Q2 into a Hamilton cycle C2 , where C2 ∩ G is also 2-balanced. We can continue this way to find C3 , . . . , Ck . Indeed, suppose that we have found C1 , . . . , Ci for i < k. Then we can still apply Lemma 4.10 since δ(F ) − 2i ≥ δ(F ) − 2k ≥ n/3. Moreover, Cj ∩ G is 2-balanced for all j ≤ i, so (C1 ∪ · · · ∪ Ci ) ∩ G is 2ibalanced. This in turn means that Proposition 4.3 (applied with C1 ∪ · · ·∪ Ci playing the role of H) implies that after removing C1 , . . . , Ci , we still have an (ε, ε′ , D − 2i)pre-framework and can find Ci+1 . We can now put everything together to find a set of localized balanced exceptional systems and a set of Hamilton cycles which altogether cover all edges of G outside G[A, B]. The localized balanced exceptional systems will be extended to Hamilton cycles later on. Corollary 5.11. Let 0 < 1/n ≪ ε ≪ ε0 ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ 1. Suppose that (G, A, A0 , B, B0 ) is an (ε, ε′ , K, D)-framework with |G| = n and such that D ≥ n/200 and D is even. Let P := {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } be a (K, m, ε, ε1 , ε2 )-partition for G. Suppose that tK := (1 − 20ε4 )D/2K 4 ∈ N and let k := 10ε4 D. Suppose that F is a graph on V (G) such that G ⊆ F , δ(F ) ≥ 2n/5 and such that F satisfies (WF5) with respect to ε′ . Then there are k edge-disjoint Hamilton cycles C1 , . . . , Ck in F and for all 1 ≤ i1 , i2 , i3 , i4 ≤ K there is a set Ji1 i2 i3 i4 such that the following properties are satisfied: (i) Ji1 i2 i3 i4 consists of tK (i1 , i2 , i3 , i4 )-BES in G with respect to P and with parameter ε0 which are edge-disjoint from each other and from C1 ∪ · · · ∪ Ck . (ii) Whenever (i1 , i2 , i3 , i4 ) 6= (i′1 , i′2 , i′3 , i′4 ), J ∈ Ji1 i2 i3 i4 and J ′ ∈ Ji′1 i′2 i′3 i′4 , then J and J ′ are edge-disjoint. (iii) Given any i ≤ k and v ∈ A0 ∪ B0 , the two edges incident to v in Ci lie in G. (iv) Let G⋄ be the subgraph of G obtained by deleting the edges of all the Ci and all the balanced exceptional systems in Ji1 i2 i3 i4 (for all 1 ≤ i1 , i2 , i3 , i4 ≤ K). Then G⋄ is bipartite with vertex classes A′ , B ′ and V0 = A0 ∪B0 is an isolated set in G⋄ .

Proof. This follows immediately from Lemmas 5.9 and 5.10(b). Indeed, clearly (i)– (iii) are satisfied. To check (iv), note that G⋄ is obtained from the graph G∗ defined in Lemma 5.10 by deleting all the edges of the Hamilton cycles Ci . But Lemma 5.10


implies that the Ci together cover all the edges in G∗ − G∗ [A, B]. Thus this implies that G⋄ is bipartite with vertex classes A′ , B ′ and V0 is an isolated set in G⋄ . 6. Special factors and balanced exceptional factors As discussed in the proof sketch, the proof of Theorem 1.5 proceeds as follows. First we find an approximate decomposition of the given graph G and finally we find a decomposition of the (sparse) leftover from the approximate decomposition (with the aid of a ‘robustly decomposable’ graph we removed earlier). Both the approximate decomposition as well as the actual decomposition steps assume that we work with a bipartite graph on A ∪ B (with |A| = |B|). So in both steps, we would need A0 ∪ B0 to be empty, which we clearly cannot assume. On the other hand, in both steps, one can specify ‘balanced exceptional path systems’ (BEPS) in G with the following crucial property: one can replace each BEPS with a path system BEPS∗ so that (α1 ) BEPS∗ is bipartite with vertex classes A and B; (α2 ) a Hamilton cycle C ∗ in G∗ := G[A, B] + BEPS∗ which contains BEPS∗ corresponds to a Hamilton cycle C in G which contains BEPS (see Section 6.1). Each BEPS will contain one of the balanced exceptional sequences BES constructed in Section 5. BEPS∗ will then be obtained by replacing the edges in BES by suitable ‘fictive’ edges (i.e. which are not necessarily contained in G). So, roughly speaking, this allows us to work with G∗ rather than G in the two steps. A convenient way of specifying and handling these balanced exceptional path systems is to combine them into ‘balanced exceptional factors’ BF (see Section 6.3 for the definition). One complication is that the ‘robust decomposition lemma’ (Lemma 7.4) we use from [15] deals with digraphs rather than undirected graphs. So to be able to apply it, we need a suitable orientation of the edges of G and so we will actually consider directed path systems BEPS∗dir instead of BEPS∗ above (whereas the path systems BEPS are undirected). The formulation of the robust decomposition lemma is quite general and rather than guaranteeing (α2 ) directly, it assumes the existence of certain directed ‘special paths systems’ SPS which are combined into ‘special factors’ SF. These are introduced in Section 6.2. Each of the Hamilton cycles produced by the lemma then contains exactly one of these special path systems. So to apply the lemma, it suffices to check separately that each BEPS∗dir satisfies the conditions required of a special path system and that it also satisfies (α2 ). 6.1. Constructing the graphs J ∗ from the balanced exceptional systems J. Suppose that J is a balanced exceptional system in a graph G with respect to a (K, m, ε0 )-partition P = {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } of V (G). We will now use J to define an auxiliary matching J ∗ . Every edge of J ∗ will have one endvertex in A and its other endvertex in B. We will regard J ∗ as being edge-disjoint from the original graph G. So even if both J ∗ and G have an edge between the same pair of endvertices, we will regard these as different edges. The edges of such a J ∗


will be called fictive edges. Proposition 6.1(ii) below shows that a Hamilton cycle in G[A ∪ B] + J ∗ containing all edges of J ∗ in a suitable order will correspond to a Hamilton cycle in G which contains J. So when finding our Hamilton cycles, this property will enable us to ignore all the vertices in V0 = A0 ∪ B0 and to consider a bipartite (multi-)graph between A and B instead. ∗ on A∪B and We construct J ∗ in two steps. First we will construct a matching JAB ∗ then J . Since each maximal path in J has endpoints in A ∪ B and internal vertices ∗ in V0 by (BES1), a balanced exceptional system J naturally induces a matching JAB on A ∪ B. More precisely, if P1 , . . . , Pℓ′ are the non-trivial paths in J and xi , yi are ∗ := {x y : i ≤ ℓ′ }. Thus J ∗ the endpoints of Pi , then we define JAB i i AB is a matching ∗ ∗ and E(J) cover exactly the same verby (BES1) and e(JAB ) ≤ e(J). Moreover, JAB tices in A. Similarly, they cover exactly the same vertices in B. So (BES3) implies ∗ [A]) = e(J ∗ [B]). We can write E(J ∗ [A]) = {x x , . . . , x that e(JAB 1 2 2s−1 x2s }, AB AB ∗ ∗ [A, B]) = {x E(JAB [B]) = {y1 y2 , . . . , y2s−1 y2s } and E(JAB y , . . . , xs′ ys′ }, 2s+1 2s+1 where xi ∈ A and yi ∈ B. Define J ∗ := {xi yi : 1 ≤ i ≤ s′ }. Note that ∗ ) ≤ e(J). All edges of J ∗ are called fictive edges. e(J ∗ ) = e(JAB As mentioned before, we regard J ∗ as being edge-disjoint from the original graph G. Suppose that P is an orientation of a subpath of (the multigraph) G[A ∪ B] + J ∗ . We say that P is consistent with J ∗ if P contains all the edges of J ∗ and P traverses the vertices x1 , y1 , x2 , . . . , ys′ −1 , xs′ , ys′ in this order. (This ordering will be crucial for the vertices x1 , y1 , . . . , x2s , y2s , but it is also convenient to have an ordering involving all vertices of J ∗ .) Similarly, we say that a cycle D in G[A∪B]+J ∗ is consistent with J ∗ if D contains all the edges of J ∗ and there exists some orientation of D which traverses the vertices x1 , y1 , x2 , . . . , ys′ −1 , xs′ , ys′ in this order. The next result shows that if J is a balanced exceptional system and C is a Hamilton cycle on A ∪ B which is consistent with J ∗ , then the graph obtained from C by replacing J ∗ with J is a Hamilton cycle on V (G) which contains J, see Figure 2. When choosing our Hamilton cycles, this property will enable us ignore all the vertices in V0 and edges in A and B and to consider the (almost complete) bipartite graph with vertex classes A and B instead. Proposition 6.1. Let P = {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } be a (K, m, ε)-partition of a vertex set V . Let G be a graph on V and let J be a balanced exceptional system with respect to P. (i) Assume that P is an orientation of a subpath of G[A ∪ B] + J ∗ such that P is consistent with J ∗ . Then the graph obtained from P − J ∗ + J by ignoring the orientations of the edges is a path on V (P ) ∪ V0 whose endvertices are the same as those of P . (ii) If J ⊆ G and D is a Hamilton cycle of G[A ∪ B] + J ∗ which is consistent with J ∗ , then D − J ∗ + J is a Hamilton cycle of G.

∗ [A]) = e(J ∗ [B]) and J ⋄ := {x y , . . . , x y } Proof. We first prove (i). Let s := e(JAB 1 1 2s 2s AB (where the xi and yi are as in the definition of J ∗ ). So J ∗ := J ⋄ ∪{x2s+1 y2s+1 , . . . , xs′ ys′ }, where s′ := e(J ∗ ). Let P c denote the path obtained from P = z1 . . . z2 by reversing its direction. (So P c = z2 . . . z1 traverses the vertices ys′ , xs′ , y2s′ −1 , . . . , x2 , y1 , x1 in


A0

B0

A0

A

B

A

(a) J

(b)

∗ JAB

B0

A0

B

A

B0

B (c) J

∗

∗ Figure 2. The thick lines illustrate the edges of J, JAB and J ∗ respectively.

this order.) First note P ′ := z1 P x1 x2 P c y1 y2 P x3 x4 P c y3 y4 . . . x2s−1 x2s P c y2s−1 y2s P z2 is a path on V (P ). Moreover, the underlying undirected graph of P ′ is precisely ∗ ∗ ∗ P − J ⋄ + (JAB [A] ∪ JAB [B]) = P − J ∗ + JAB .

∗ . Now recall that if w w is an edge in J ∗ , then In particular, P ′ contains JAB 1 2 AB the vertices w1 and w2 are the endpoints of some path P ∗ in J (where the internal vertices on P ∗ lie in V0 ). Clearly, P ′ − w1 w2 + P ∗ is also a path. Repeating this step ∗ gives a path P ′′ on V (P )∪V . Moreover, P ′′ = P −J ∗ +J. for every edge w1 w2 of JAB 0 This completes the proof of (i). (ii) now follows immediately from (i).

6.2. Special path systems and special factors. As mentioned earlier, in order to apply Lemma 7.4, we first need to prove the existence of certain ‘special path systems’. These are defined below. Suppose that P = {A0 , A1 , . . . , AK , B0 , B1 , . . . , BK } is a (K, m, ε0 )-partition of a vertex set V and L, m/L ∈ N. We say that (P, P ′ ) is a (K, L, m, ε0 )-partition of V if P ′ is obtained from P by partitioning Ai into L sets Ai,1 , . . . , Ai,L of size m/L for all 1 ≤ i ≤ K and partitioning Bi into L sets Bi,1 , . . . , Bi,L of size m/L for all 1 ≤ i ≤ K. (So P ′ consists of the exceptional sets A0 , B0 , the KL clusters Ai,j and the KL clusters Bi,j .) Unless stated otherwise, whenever considering a (K, L, m, ε0 )-partition (P, P ′ ) of a vertex set V we use the above notation to denote the elements of P and P ′ . Let (P, P ′ ) be a (K, L, m, ε0 )-partition of V . Consider a spanning cycle C = A1 B1 . . . AK BK on the clusters of P. Given an integer f dividing K, the canonical interval partition I of C into f intervals consists of the intervals A(i−1)K/f +1 B(i−1)K/f +1 A(i−1)K/f +2 . . . BiK/f AiK/f +1


for all i ≤ f . (Here AK+1 := A1 .) Suppose that G is a digraph on V \ V0 and h ≤ L. Let I = Aj Bj Aj+1 . . . Aj ′ be an interval in I. A special path system SP S of style h in G spanning the interval I consists of precisely m/L (non-trivial) vertex-disjoint directed paths P1 , . . . , Pm/L such that the following conditions hold: (SPS1) Every Ps has its initial vertex in Aj,h and its final vertex in Aj ′ ,h . (SPS2) SP S contains a matching Fict(SP S) such that all the edges in Fict(SP S) avoid the endclusters Aj and Aj ′ of I and such that E(Ps ) \ Fict(SP S) ⊆ E(G). (SPS3) The vertex set of SP S is Aj,h ∪ Bj,h ∪ Aj+1,h ∪ · · · ∪ Bj ′ −1,h ∪ Aj ′ ,h .

The edges in Fict(SP S) are called fictive edges of SP S. Let I = {I1 , . . . , If } be the canonical interval partition of C into f intervals. A special factor SF with parameters (L, f ) in G (with respect to C, P ′ ) is a 1-regular digraph on V \ V0 which is the union of Lf digraphs SP Sj,h (one for all j ≤ f and h ≤ L) such that each SP Sj,h is a special path system of style h in G which spans Ij . We write Fict(SF ) for the union of the sets Fict(SP Sj,h ) over all j ≤ f and h ≤ L and call the edges in Fict(SF ) fictive edges of SF . We will always view fictive edges as being distinct from each other and from the edges in other digraphs. So if we say that special factors SF1 , . . . , SFr are pairwise edge-disjoint from each other and from some digraph Q on V \ V0 , then this means that Q and all the SFi − Fict(SFi ) are pairwise edge-disjoint, but for example there could be an edge from x to y in Q as well as in Fict(SFi ) for several indices i ≤ r. But these are the only instances of multiedges that we allow, i.e. if there is more than one edge from x to y, then all but at most one of these edges are fictive edges. 6.3. Balanced exceptional path systems and balanced exceptional factors. We now define balanced exceptional path systems BEPS. It will turn out that they (or rather their bipartite directed versions BEPS∗dir involving fictive edges) will satisfy the conditions of the special path systems defined above. Moreover, (bipartite) Hamilton cycles containing BEPS∗dir correspond to Hamilton cycles in the ‘original’ graph G (see Proposition 6.2). Let (P, P ′ ) be a (K, L, m, ε0 )-partition of a vertex set V . Suppose that K/f ∈ N and h ≤ L. Consider a spanning cycle C = A1 B1 . . . AK BK on the clusters of P. Let I be the canonical interval partition of C into f intervals of equal size. Suppose that G is an oriented bipartite graph with vertex classes A and B. Suppose that I = Aj Bj . . . Aj ′ is an interval in I. A balanced exceptional path system BEP S of style h for G spanning I consists of precisely m/L (non-trivial) vertex-disjoint undirected paths P1 , . . . , Pm/L such that the following conditions hold:

(BEPS1) Every Ps has one endvertex in Aj,h and its other endvertex in Aj ′ ,h . (BEPS2) J := BEP S − BEP S[A, B] is a balanced exceptional system with respect to P such that P1 contains all edges of J and so that the edge set of J is disjoint from Aj,h and Aj ′ ,h . Let P1,dir be the path obtained by orienting P1 towards its endvertex in Aj ′ ,h and let Jdir be the orientation of J obtained in ∗ be obtained from J ∗ by orienting every edge in this way. Moreover, let Jdir

´ ¨ 42 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN ∗ is a directed ∗ := P1,dir − Jdir + Jdir J ∗ towards its endvertex in B. Then P1,dir ∗ path from Aj,h to Aj ′ ,h which is consistent with J . (BEPS3) The vertex set of BEP S is V0 ∪ Aj,h ∪ Bj,h ∪ Aj+1,h ∪ · · · ∪ Bj ′ −1,h ∪ Aj ′ ,h . (BEPS4) For each 2 ≤ s ≤ m/L, define Ps,dir similarly as P1,dir . Then E(Ps,dir ) \ E(Jdir ) ⊆ E(G) for every 1 ≤ s ≤ m/L.

∗ be the path system consisting of P ∗ , P Let BEP Sdir 2,dir , . . . , Pm/L,dir . Then 1,dir ∗ BEP Sdir is a special path system of style h in G which spans the interval I and such ∗ ) = J∗ . that Fict(BEP Sdir dir Let I = {I1 , . . . , If } be the canonical interval partition of C into f intervals. A balanced exceptional factor BF with parameters (L, f ) for G (with respect to C, P ′ ) is the union of Lf undirected graphs BEP Sj,h (one for all j ≤ f and h ≤ L) such that each BEP Sj,h is a balanced exceptional path system of style h for G which ∗ for the union of BEP S ∗ spans Ij . We write BFdir j,h,dir over all j ≤ f and h ≤ L. ∗ Note that BFdir is a special factor with parameters (L, f ) in G (with respect to C, ∗ ) is the union of J ∗ P ′ ) such that Fict(BFdir j,h,dir over all j ≤ f and h ≤ L, where Jj,h = BEP Sj,h − BEP Sj,h [A, B] is the balanced exceptional system contained in ∗ is a 1-regular digraph on BEP Sj,h (see condition (BEPS2)). In particular, BFdir V \ V0 while BF is an undirected graph on V with

(6.1)

dBF (v) = 2 for all v ∈ V \ V0

and

dBF (v) = 2Lf for all v ∈ V0 .

Given a balanced exceptional path system BEP S, let J be as in (BEPS2) and let BEP S ∗ := BEP S − J + J ∗ . So BEP S ∗ consists of P1∗ := P1 − J + J ∗ as well as P2 , . . . , Pm/L . The following is an immediate consequence of (BEPS2) and Proposition 6.1. Proposition 6.2. Let (P, P ′ ) be a (K, L, m, ε0 )-partition of a vertex set V . Suppose that G is a graph on V \ V0 , that Gdir is an orientation of G[A, B] and that BEP S is a balanced exceptional path system for Gdir . Let J be as in (BEPS2). Let C be a Hamilton cycle of G + J ∗ which contains BEP S ∗ . Then C − BEP S ∗ + BEP S is a Hamilton cycle of G ∪ J. Proof. Note that C − BEP S ∗ + BEP S = C − J ∗ + J. Moreover, (BEPS2) implies that C contains all edges of J ∗ and is consistent with J ∗ . So the proposition follows from Proposition 6.1(ii) applied with G ∪ J playing the role of G. 6.4. Finding balanced exceptional factors in a scheme. The following definition of a ‘scheme’ captures the ‘non-exceptional’ part of the graphs we are working with. For example, this will be the structure within which we find the edges needed to extend a balanced exceptional system into a balanced exceptional path system. Given an oriented graph G and partitions P and P ′ of a vertex set V , we call (G, P, P ′ ) a [K, L, m, ε0 , ε]-scheme if the following properties hold:

(Sch1′ ) (P, P ′ ) is a (K, L, m, ε0 )-partition of V . Moreover, V (G) = A ∪ B. (Sch2′ ) Every edge of G has one endvertex in A and its other endvertex in B.


(Sch3′ ) G[Ai,j , Bi′ ,j ′ ] and G[Bi′ ,j ′ , Ai,j ] are [ε, 1/2]-superregular for all i, i′ ≤ K and all j, j ′ ≤ L. Further, G[Ai , Bj ] and G[Bj , Ai ] are [ε, 1/2]-superregular for all i, j ≤ K. ′ (Sch4 ) |NG+ (x) ∩ NG− (y) ∩ Bi,j | ≥ (1 − ε)m/5L for all distinct x, y ∈ A, all i ≤ K and all j ≤ L. Similarly, |NG+ (x) ∩ NG− (y) ∩ Ai,j | ≥ (1 − ε)m/5L for all distinct x, y ∈ B, all i ≤ K and all j ≤ L. If L = 1 (and so P = P ′ ), then (Sch1′ ) just says that P is a (K, m, ε0 )-partition of V (G). The next lemma allows us to extend a suitable balanced exceptional system into a balanced exceptional path system. Given h ≤ L, we say that an (i1 , i2 , i3 , i4 )-BES J has style h (with respect to the (K, L, m, ε0 )-partition (P, P ′ )) if all the edges of J have their endvertices in V0 ∪ Ai1 ,h ∪ Ai2 ,h ∪ Bi3 ,h ∪ Bi4 ,h . Lemma 6.3. Suppose that K, L, n, m/L ∈ N, that 0 < 1/n ≪ ε, ε0 ≪ 1 and ε0 ≪ 1/K, 1/L. Let (G, P, P ′ ) be a [K, L, m, ε0 , ε]-scheme with |V (G) ∪ V0 | = n. Consider a spanning cycle C = A1 B1 . . . AK BK on the clusters of P and let I = Aj Bj Aj+1 . . . Aj ′ be an interval on C of length at least 10. Let J be an (i1 , i2 , i3 , i4 )-BES of style h ≤ L with parameter ε0 (with respect to (P, P ′ )), for some i1 , i2 , i3 , i4 ∈ {j + 1, . . . , j ′ − 1}. Then there exists a balanced exceptional path system of style h for G which spans the interval I and contains all edges in J. Proof. For each k ≤ 4, let mk denote the number of vertices in Aik ,h ∪ Bik ,h which are incident to edges of J. We only consider the case when i1 , i2 , i3 and i4 are distinct and mk > 0 for each k ≤ 4, as the other cases can be proved by similar arguments. Clearly m1 + · · · + m4 ≤ 2ε0 n by (BES4). For every vertex x ∈ A, we define B(x) to be the cluster Bi,h ∈ P ′ such that Ai contains x. Similarly, for every y ∈ B, we define A(y) to be the cluster Ai,h ∈ P ′ such that Bi contains y. Let x1 y1 , . . . , xs′ ys′ be the edges of J ∗ , with xi ∈ A and yi ∈ B for all i ≤ s′ . (Recall that the ordering of these edges is fixed in the definition of J ∗ .) Thus s′ = (m1 +· · ·+m4 )/2 ≤ ε0 n. Moreover, our assumption that ε0 ≪ 1/K, 1/L implies that ε0 n ≤ m/100L (say). Together with (Sch4′ ) this in turn ensures that for every r ≤ s′ , we can pick vertices wr ∈ B(xr ) and zr ∈ A(yr ) such that wr xr , yr zr and zr wr+1 are (directed) edges in G and such that all the 4s′ vertices xr , yr , wr , zr (for r ≤ s′ ) are distinct from each other. Let P1′ be the path w1 x1 y1 z1 w2 x2 y2 z2 w3 . . . ys′ zs′ . Thus ∗ which is consistent with J ∗ . (Here J ∗ P1′ is a directed path from B to A in G + Jdir dir is obtained from J ∗ by orienting every edge towards B.) Note that |V (P1′ ) ∩ Aik ,h | = mk = |V (P1′ ) ∩ Bik ,h | for all k ≤ 4. (This follows from our assumption that i1 , i2 , i3 / {i1 , i2 , i3 , i4 }. and i4 are distinct.) Moreover, V (P1′ ) ∩ (Ai ∪ Bi ) = ∅ for all i ∈ Pick a vertex z ′ in Aj,h so that z ′ w1 is an edge of G. Find a path P1′′ from zs′ to Aj ′ ,h in G such that the vertex set of P1′′ consists of zs′ and precisely one vertex in each Ai,h for all i ∈ {j + 1, . . . , j ′ } \ {i1 , i2 , i3 , i4 } and one vertex in each Bi,h for all i ∈ {j, . . . , j ′ − 1} \ {i1 , i2 , i3 , i4 } and no other vertices. (Sch4′ ) ensures that this can ∗ to be the concatenation of z ′ w1 , P1′ and P1′′ . Note be done greedily. Define P1,dir ∗ which is consistent with ∗ is a directed path from Aj,h to Aj ′ ,h in G + Jdir that P1,dir

´ ¨ 44 BELA CSABA, DANIELA KUHN, ALLAN LO, DERYK OSTHUS AND ANDREW TREGLOWN ∗ )⊆ J ∗ . Moreover, V (P1,dir

while

S

Ai,h ∪ Bi,h ,   for i ∈ {j, . . . , j ′ } \ {i1 , i2 , i3 , i4 }, 1 ∗ |V (P1,dir ) ∩ Ai,h | = mk for i = ik and k ≤ 4,   0 otherwise, i≤K

  1 ∗ |V (P1,dir ) ∩ Bi,h | = mk   0

for i ∈ {j, . . . , j ′ − 1} \ {i1 , i2 , i3 , i4 }, for i = ik and k ≤ 4, otherwise.

k (Sch4′ ) ensures that for each k ≤ 4, there exist mk −1 (directed) paths P1k , . . . , Pm k −1 in G such that • Prk is a path from Aj,h to Aj ′ ,h for each r ≤ mk − 1 and k ≤ 4; • each Prk contains precisely one vertex in Ai,h for each i ∈ {j, . . . , j ′ } \ {ik }, one vertex in Bi,h for each i ∈ {j, . . . , j ′ − 1} \ {ik } and no other vertices; 4 2 ∗ , P 1, . . . , P 1 • P1,dir m1 −1 , P1 , . . . , Pm4 −1 are vertex-disjoint. 1

∗ Let Q be the union of P1,dir and all the Prk over all k ≤ 4 and r ≤ mk − 1. Thus Q is a path system consisting of m1 + · · · + m4 − 3 vertex-disjoint directed paths from Aj,h to Aj ′ ,h . Moreover, V (Q) consists of precisely m1 + · · · + m4 − 3 ≤ 2ε0 n vertices in Ai,h for every j ≤ i ≤ j ′ and precisely m1 + · · · + m4 − 3 vertices in Bi,h for every ′ := B j ≤ i < j ′ . Set A′i,h := Ai,h \ V (Q) and Bi,h i,h \ V (Q) for all i ≤ K. Note that, ′ for all j ≤ i ≤ j , √ m m m m (6.2) |A′i,h | = − (m1 + · · · + m4 − 3) ≥ − 2ε0 n ≥ − 5ε0 mK ≥ (1 − ε0 ) L L L L ′ | ≥ (1 − √ε )m/L for all j ≤ i < j ′ . Pick since ε0 ≪ 1/K, 1/L. Similarly, |Bi,h 0 a new constant ε′ such that ε, ε0 ≪ ε′ ≪ 1. Then (Sch3′ ) and (6.2) together with ′ ] is still [ε′ , 1/2]-superregular and so we can Proposition 2.1 imply that G[A′i,h , Bi,h ′ ] for all j ≤ i < j ′ . Similarly, we can find find a perfect matching in G[A′i,h , Bi,h ′ ′ a perfect matching in G[Bi,h , Ai+1,h ] for all j ≤ i < j ′ . The union Q′ of all these matchings forms m/L − (m1 + · · · + m4 ) + 3 vertex-disjoint directed paths. ∗ + J by ignoring the ∗ − Jdir Let P1 be the undirected graph obtained from P1,dir ∗ )∪V directions of all the edges. Proposition 6.1(i) implies that P1 is a path on V (P1,dir 0 ∗ with the same endvertices as P1,dir . Consider the path system obtained from (Q ∪ ∗ } by ignoring the directions of the edges on all the paths. Let BEP S be Q′ ) \ {P1,dir the union of this path system and P1 . Then BEP S is a balanced exceptional path system for G, as required.

The next lemma shows that we can obtain many edge-disjoint balanced exceptional factors by extending balanced exceptional systems with suitable properties. Lemma 6.4. Suppose that L, f, q, n, m/L, K/f ∈ N, that K/f ≥ 10, that 0 < 1/n ≪ ε, ε0 ≪ 1, that ε0 ≪ 1/K, 1/L and Lq/m ≪ 1. Let (G, P, P ′ ) be a [K, L, m, ε0 , ε]scheme with |V (G) ∪ V0 | = n. Consider a spanning cycle C = A1 B1 . . . AK BK on


the clusters of P. Suppose that there exists a set J of Lf q edge-disjoint balanced exceptional systems with parameter ε0 such that • for all i ≤ f and all h ≤ L, J contains precisely q (i1 , i2 , i3 , i4 )-BES of style h (with respect to (P, P ′ )) for which i1 , i2 , i3 , i4 ∈ {(i−1)K/f +2, . . . , iK/f }.

Then there exist q edge-disjoint balanced exceptional S factors with parameters (L, f ) for G (with respect to C, P ′ ) covering all edges in J .

Recall that the canonical interval partition I of C into f intervals consists of the intervals A(i−1)K/f +1 B(i−1)K/f +1 A(i−1)K/f +2 . . . AiK/f +1

for all i ≤ f . So the condition on J ensures that for each interval I ∈ I and each h ≤ L, the set J contains precisely q balanced exceptional systems of style h whose edges are only incident to vertices in V0 and vertices belonging to clusters in the interior of I. We will use Lemma 6.3 to extend each such balanced exceptional system into a balanced exceptional path system of style h spanning I. Proof of Lemma 6.4. Choose a new constant ε′ with ε, Lq/m ≪ ε′ ≪ 1. Let J1 , . . . , Jq be a partition of J such that for all j ≤ q, h ≤ L and i ≤ f , the set Jj contains precisely one (i1 , i2 , i3 , i4 )-BES of style h with i1 , i2 , i3 , i4 ∈ {(i − 1)K/f + 2, . . . , iK/f }. Thus each Jj consists of Lf balanced exceptional systems. For each j ≤ q in turn, we will choose a balanced exceptional factor EFj with parameters (L, f ) for G such that BFj and BFj ′ are edge-disjoint for all j ′ < j and BFj contains all edges of the balanced exceptional systems in Jj . Assume that we have already constructed BF1 , . . . , BFj−1 . In order to construct BFj , we will choose the Lf balanced exceptional path systems forming BFj one by one, such that each of these balanced exceptional path systems is edge-disjoint from BF1 , . . . , BFj−1 and contains precisely one of the balanced exceptional systems in Jj . Suppose that we have already chosen some of these balanced exceptional path systems and that next we wish to choose a balanced exceptional path system of style h which spans the interval I ∈ I of C and contains J ∈ Jj . Let G′ be the oriented graph obtained from G by deleting all the edges in the balanced path systems already chosen for BFj as well as deleting all the edges in BF1 , . . . , BFj−1 . Recall from (Sch1′ ) that V (G) = A∪ B. Thus ∆(G− G′ ) ≤ 2j < 3q by (6.1). Together with Proposition 2.1 this implies that (G′ , P, P ′ ) is still a [K, L, m, ε0 , ε′ ]-scheme. (Here we use that ∆(G − G′ ) < 3q = 3Lq/m · m/L and ε, Lq/m ≪ ε′ ≪ 1.) So we can apply Lemma 6.3 with ε′ playing the role of ε to obtain a balanced exceptional path system of style h for G′ (and thus for G) which spans I and contains all edges of J. This completes the proof of the lemma.

7. The robust decomposition lemma The robust decomposition lemma (Corollary 7.5) allows us to transform an approximate Hamilton decomposition into an exact one. As discussed in Section 3, it will only be used in the proof of Theorem 1.5 (and not in the proof of Theorem 1.6).


In the next subsection, we introduce the necessary concepts. In particular, Corollary 7.5 relies on the existence of a so-called bi-universal walk. The (proof of the) robust decomposition lemma then uses edges guaranteed by this universal walk to ‘balance out’ edges of the graph H when constructing the Hamilton decomposition of Grob + H. 7.1. Chord sequences and bi-universal walks. Let R be a digraph whose vertices are V1 , . . . , Vk and suppose that C = V1 . . . Vk is a Hamilton cycle of R. (Later on the vertices of R will be clusters. So we denote them by capital letters.) A chord sequence CS(Vi , Vj ) from Vi to Vj in R is an ordered sequence of edges of the form CS(Vi , Vj ) = (Vi1 −1 Vi2 , Vi2 −1 Vi3 , . . . , Vit −1 Vit+1 ), where Vi1 = Vi , Vit+1 = Vj and the edge Vis −1 Vis+1 belongs to R for each s ≤ t. If i = j then we consider the empty set to be a chord sequence from Vi to Vj . Without loss of generality, we may assume that CS(Vi , Vj ) does not contain any edges of C. (Indeed, suppose that Vis −1 Vis+1 is an edge of C. Then is = is+1 and so we can obtain a chord sequence from Vi to Vj with fewer edges.) For example, if Vi−1 Vi+2 ∈ E(R), then the edge Vi−1 Vi+2 is a chord sequence from Vi to Vi+2 . The crucial property of chord sequences is that they satisfy a ‘local balance’ condition. Suppose that CS is obtained by concatenating several chord sequences CS(Vi1 , Vi2 ), CS(Vi2 , Vi3 ), . . . , CS(Viℓ−1 , Viℓ ), CS(Viℓ , Viℓ+1 ) where Vi1 = Viℓ+1 . Then for every Vi , the number of edges of CS leaving Vi−1 equals the number of edges entering Vi . We will not use this property explicitly, but it underlies the proofs of e.g. Lemma 7.4 and appears implicitly e.g. in (BU3) below. A closed walk U in R is a bi-universal walk for C with parameter ℓ′ if the following conditions hold: (BU1) The edge set of U has a partition into Uodd and Ueven . For every 1 ≤ i ≤ k there is a chord sequence ECS bi (Vi , Vi+2 ) from Vi to Vi+2 such that Ueven contains all edges of all these chord sequences for even i (counted with multiplicities) and Uodd contains all edges of these chord sequences for odd i. All remaining edges of U lie on C. √ (BU2) Each ECS bi (Vi , Vi+2 ) consists of at most ℓ′ /2 edges. (BU3) Ueven enters every cluster Vi exactly ℓ′ /2 times and it leaves every cluster Vi exactly ℓ′ /2 times. The same assertion holds for Uodd . Note that condition (BU1) means that if an edge Vi Vj ∈ E(R) \ E(C) occurs in total 5 times (say) in ECS bi (V1 , V3 ), . . . , ECS bi (Vk , V2 ) then it occurs precisely 5 times in U . We will identify each occurrence of Vi Vj in ECS bi (V1 , V3 ), . . . , ECS bi (Vk , V2 ) with a (different) occurrence of Vi Vj in U . Note that the edges of ECS bi (Vi , Vi+2 ) are allowed to appear in a different order within U . Lemma 7.1. Let R be a digraph with vertices V1 , . . . , Vk where k ≥ 4 is even. Suppose that C = V1 . . . Vk is a Hamilton cycle of R and that Vi−1 Vi+2 ∈ E(R) for every 1 ≤ i ≤ k. Let ℓ′ ≥ 4 be an even integer. Let Ubi,ℓ′ denote the multiset obtained from ℓ′ − 1 copies of E(C) by adding Vi−1 Vi+2 ∈ E(R) for every 1 ≤ i ≤ k. Then


the edges in Ubi,ℓ′ can be ordered so that the resulting sequence forms a bi-universal walk for C with parameter ℓ′ . In the remainder of the paper, we will also write Ubi,ℓ′ for the bi-universal walk guaranteed by Lemma 7.1. Proof. Let us first show that the edges in Ubi,ℓ′ can be ordered so that the resulting sequence forms a closed walk in R. To see this, consider the multidigraph U obtained from Ubi,ℓ′ by deleting one copy of E(C). Then U is (ℓ′ − 1)-regular and thus has a decomposition into 1-factors. We order the edges of Ubi,ℓ′ as follows: We first traverse all cycles of the 1-factor decomposition of U which contain the cluster V1 . Next, we traverse the edge V1 V2 of C. Next we traverse all those cycles of the 1-factor decomposition which contain V2 and which have not been traversed so far. Next we traverse the edge V2 V3 of C and so on until we reach V1 again. Recall that, for each 1 ≤ i ≤ k, the edge Vi−1 Vi+2 is a chord sequence from Vi to Vi+2 . Thus we can take ECS bi (Vi , Vi+2 ) := Vi−1 Vi+2 . Then Ubi,ℓ′ satisfies (BU1)– (BU3). Indeed, (BU2) is clearly satisfied. Partition one of the copies of E(C) in Ubi,ℓ′ into Eeven and Eodd where Eeven = {Vi Vi+1 | i even} and Eodd = {Vi Vi+1 | i odd}. Note that the union of Eeven together with all ECS bi (Vi , Vi+2 ) for even i is a 1factor in R. Add ℓ′ /2 − 1 of the remaining copies of E(C) to this 1-factor to obtain Ueven . Define Uodd to be E(Ubi,ℓ′ ) \ Ueven . By construction of Ueven and Uodd , (BU1) and (BU3) are satisfied.

7.2. Bi-setups and the robust decomposition lemma. The aim of this subsection is to state the robust decomposition lemma (Lemma 7.4, proved in [15]) and derive Corollary 7.5, which we shall use later on. The robust decomposition lemma guarantees the existence of a ‘robustly decomposable’ digraph Grob dir within a ‘bi-setup’. Roughly speaking, a bi-setup is a digraph G together with its ‘reduced digraph’ R, which contains a Hamilton cycle C and a universal walk U . In our application, G[A, B] will play the role of G and R will be the complete bipartite digraph. To define a bi-setup formally, we first need to define certain ‘refinements’ of partitions. Given a digraph G and a partition P of V (G) into k clusters V1 , . . . , Vk of equal size, we say that a partition P ′ of V (G) is an ℓ′ -refinement of P if P ′ is obtained by splitting each Vi into ℓ′ subclusters of equal size. (So P ′ consists of ℓ′ k clusters.) P ′ is an ε-uniform ℓ′ -refinement of P if it is an ℓ′ -refinement of P which satisfies the following condition: Whenever x is a vertex of G, V is a cluster in P and |NG+ (x) ∩ V | ≥ ε|V | then |NG+ (x) ∩ V ′ | = (1 ± ε)|NG+ (x) ∩ V |/ℓ′ for each cluster V ′ ∈ P ′ with V ′ ⊆ V . The inneighbourhoods of the vertices of G satisfy an analogous condition. We will use the following lemma from [15]. Lemma 7.2. Suppose that 0 < 1/m ≪ 1/k, ε ≪ ε′ , d, 1/ℓ ≤ 1 and that k, ℓ, m/ℓ ∈ N. Suppose that G is a digraph and that P is a partition of V (G) into k clusters of size m. Then there exists an ε-uniform ℓ-refinement of P. Moreover, any ε-uniform ℓ-refinement P ′ of P automatically satisfies the following condition:


• Suppose that V , W are clusters in P and V ′ , W ′ are clusters in P ′ with V ′ ⊆ V and W ′ ⊆ W . If G[V, W ] is [ε, d′ ]-superregular for some d′ ≥ d then G[V ′ , W ′ ] is [ε′ , d′ ]-superregular. We will also need the following definition from [15], which describes the structure within which the robust decomposition lemma finds the robustly decomposable graph. (G, P, P ′ , R, C, U, U ′ ) is called an (ℓ′ , k, m, ε, d)-bi-setup if the following properties are satisfied: (ST1) G and R are digraphs. P is a partition of V (G) into k clusters of size m where k is even. The vertex set of R consists of these clusters. (ST2) For every edge V W of R, the corresponding pair G[V, W ] is (ε, ≥ d)-regular. (ST3) C = V1 . . . Vk is a Hamilton cycle of R and for every edge Vi Vi+1 of C the corresponding pair G[Vi , Vi+1 ] is [ε, ≥ d]-superregular. (ST4) U is a bi-universal walk for C in R with parameter ℓ′ and P ′ is an ε-uniform ℓ′ -refinement of P. ′ (ST5) Let Vj1 , . . . , Vjℓ denote the clusters in P ′ which are contained in Vj (for each 1 ≤ j ≤ k). Then U ′ is a closed walk on the clusters in P ′ which is obtained from U as follows: When U visits Vj for the ath time, we let U ′ visit the subcluster Vja (for all 1 ≤ a ≤ ℓ′ ). ′

′

(ST6) For every edge Vij Vij′ of U ′ the corresponding pair G[Vij , Vij′ ] is [ε, ≥ d]superregular. In [15], in a bi-setup, the digraph G could also contain an exceptional set, but since we are only using the definition in the case when there is no such exceptional set, we have only stated it in this special case. Suppose that (G, P, P ′ ) is a [K, L, m, ε0 , ε]-scheme and that C = A1 B1 . . . AK BK is a spanning cycle on the clusters of P. Let Pbi := {A1 , . . . , AK , B1 , . . . , BK }. ′′ be an ε-uniform ℓ′ -refinement of P Suppose that ℓ′ , m/ℓ′ ∈ N with ℓ′ ≥ 4. Let Pbi bi (which exists by Lemma 7.2). Let Cbi be the directed cycle obtained from C in which the edge A1 B1 is oriented towards B1 and so on. Let Rbi be the complete bipartite digraph whose vertex classes are {A1 , . . . , AK } and {B1 , . . . , BK }. Let Ubi,ℓ′ be a ′ bi-universal walk for C with parameter ℓ′ as defined in Lemma 7.1. Let Ubi,ℓ ′ be the closed walk obtained from Ubi,ℓ′ as described in (ST5). We will call ′ ′′ (G, Pbi , Pbi , Rbi , Cbi , Ubi,ℓ′ , Ubi,ℓ ′)

the bi-setup associated to (G, P, P ′ ). The following lemma shows that it is indeed a bi-setup. Lemma 7.3. Suppose that K, L, m/L, ℓ′ , m/ℓ′ ∈ N with ℓ′ ≥ 4, K ≥ 2 and 0 < 1/m ≪ 1/K, ε ≪ ε′ , 1/ℓ′ . Suppose that (G, P, P ′ ) is a [K, L, m, ε0 , ε]-scheme and that C = A1 B1 . . . AK BK is a spanning cycle on the clusters of P. Then ′ ′′ , Rbi , Cbi , Ubi,ℓ′ , Ubi,ℓ (G, Pbi , Pbi ′)

is an (ℓ′ , 2K, m, ε′ , 1/2)-bi-setup.

′ ′ ′′ , R , C , U Proof. Clearly, (G, Pbi , Pbi bi bi bi,ℓ′ , Ubi,ℓ′ ) satisfies (ST1). (Sch3 ) implies that (ST2) and (ST3) hold. Lemma 7.1 implies (ST4). (ST5) follows from the

PROOF OF THE 1-FACTORIZATION & HAMILTON DECOMPOSITION CONJECTURES II 49 ′ ′′ ′ definition of Ubi,ℓ ′ . Finally, (ST6) follows from (Sch3 ) and Lemma 7.2 since Pbi is an ε-uniform ℓ′ -refinement of Pbi .

We now state the robust decomposition lemma from [15]. This guarantees the existence of a ‘robustly decomposable’ digraph Grob dir , whose crucial property is that H + Grob has a Hamilton decomposition for any sparse bipartite regular digraph H dir rob which is edge-disjoint from Gdir . Grob dir consists of digraphs CAdir (r) (the ‘chord absorber’) and P CAdir (r) (the ‘parity extended cycle switcher’) together with some special factors. Grob dir is constructed in two steps: given a suitable set SF of special factors, the lemma first ‘constructs’ CAdir (r) and then, given another suitable set SF ′ of special factors, the lemma ‘constructs’ P CAdir (r). The reason for having two separate steps is that in [15], it is not clear how to construct CAdir (r) after constructing SF ′ (rather than before), as the removal of SF ′ from the digraph under consideration affects its properties considerably. Lemma 7.4. Suppose that 0 < 1/m ≪ 1/k ≪ ε ≪ 1/q ≪ 1/f ≪ r1 /m ≪ d ≪ 1/ℓ′ , 1/g ≪ 1 where ℓ′ is even and that rk 2 ≤ m. Let r2 := 96ℓ′ g2 kr,

r3 := rf k/q,

r ⋄ := r1 + r2 + r − (q − 1)r3 ,

s′ := rf k + 7r ⋄

and suppose that k/14, k/f, k/g, q/f, m/4ℓ′ , f m/q, 2f k/3g(g − 1) ∈ N. Suppose that (G, P, P ′ , R, C, U, U ′ ) is an (ℓ′ , k, m, ε, d)-bi-setup and C = V1 . . . Vk . Suppose that P ∗ is a (q/f )-refinement of P and that SF1 , . . . , SFr3 are edge-disjoint special factors with parameters (q/f, f ) with respect to C, P ∗ in G. Let SF := SF1 + · · · + SFr3 . Then there exists a digraph CAdir (r) for which the following holds: (i) CAdir (r) is an (r1 + r2 )-regular spanning subdigraph of G which is edgedisjoint from SF . (ii) Suppose that SF1′ , . . . , SFr′⋄ are special factors with parameters (1, 7) with respect to C, P in G which are edge-disjoint from each other and from CAdir (r) + SF. Let SF ′ := SF1′ + · · · + SFr′⋄ . Then there exists a digraph P CAdir (r) for which the following holds: (a) P CAdir (r) is a 5r ⋄ -regular spanning subdigraph of G which is edgedisjoint from CAdir (r) + SF + SF ′ . (b) Let SPS be the set consisting of all the s′ special path systems contained in SF +SF ′ . Let Veven denote the union of all Vi over all even 1 ≤ i ≤ k and define Vodd similarly. Suppose that H is an r-regular bipartite digraph on V (G) with vertex classes Veven and Vodd which is edge-disjoint ′ rob from Grob dir := CAdir (r) + P CAdir (r) + SF + SF . Then H + Gdir has a ′ decomposition into s edge-disjoint Hamilton cycles C1 , . . . , Cs′ . Moreover, Ci contains one of the special path systems from SPS, for each i ≤ s′ . Recall from Section 6.2 that we always view fictive edges in special factors as being distinct from each other and from the edges in other graphs. So for example, saying that CAdir (r) and SF are edge-disjoint in Lemma 7.4 still allows for a fictive edge


xy in SF to occur in CAdir (r) as well (but CAdir (r) will avoid all non-fictive edges in SF ). We will use the following ‘undirected’ consequence of Lemma 7.4. Corollary 7.5. Suppose that 0 < 1/m ≪ ε0 , 1/K ≪ ε ≪ 1/L ≪ 1/f ≪ r1 /m ≪ 1/ℓ′ , 1/g ≪ 1 where ℓ′ is even and that 4rK 2 ≤ m. Let r2 := 192ℓ′ g2 Kr,

r3 := 2rK/L,

r ⋄ := r1 + r2 + r − (Lf − 1)r3 ,

s′ := 2rf K + 7r ⋄

and suppose that L, K/7, K/f, K/g, m/4ℓ′ , m/L, 4f K/3g(g − 1) ∈ N. Suppose that (Gdir , P, P ′ ) is a [K, L, m, ε0 , ε]-scheme and let G′ denote the underlying undirected graph of Gdir . Let C = A1 B1 . . . AK BK be a spanning cycle on the clusters in P. Suppose that BF1 , . . . , BFr3 are edge-disjoint balanced exceptional factors with parameters (L, f ) for Gdir (with respect to C, P ′ ). Let BF := BF1 + · · · + BFr3 . Then there exists a graph CA(r) for which the following holds: (i) CA(r) is a 2(r1 + r2 )-regular spanning subgraph of G′ which is edge-disjoint from BF. (ii) Suppose that BF1′ , . . . , BFr′⋄ are balanced exceptional factors with parameters (1, 7) for Gdir (with respect to C, P) which are edge-disjoint from each other and from CA(r) + BF. Let BF ′ := BF1′ + · · · + BFr′⋄ . Then there exists a graph P CA(r) for which the following holds: (a) P CA(r) is a 10r ⋄ -regular spanning subgraph of G′ which is edge-disjoint from CA(r) + BF + BF ′ . (b) Let BEPS be the set consisting of all the s′ balanced exceptional path systems contained in BF +BF ′ . Suppose SK that H isSaK2r-regular bipartite graph on V (Gdir ) with vertex classes i=1 Ai and i=1 Bi which is edgedisjoint from Grob := CA(r) + P CA(r) + BF + BF ′ . Then H + Grob has a decomposition into s′ edge-disjoint Hamilton cycles C1 , . . . , Cs′ . Moreover, Ci contains one of the balanced exceptional path systems from BEPS, for each i ≤ s′ . We remark that we write A1 , . . . , AK , B1 , . . . , BK for the clusters in P. Note that the vertex set of each of EF , EF ′ , Grob includes V0 while that of Gdir , CA(r), P CA(r), H does not. Here V0 = A0 ∪ B0 , where A0 and B0 are the exceptional sets of P. Proof. Choose new constants ε′ and d such that ε ≪ ε′ ≪ 1/L and r1 /m ≪ d ≪ ′ ′′ , R , C , U 1/ℓ′ , 1/g. Consider the bi-setup (Gdir , Pbi , Pbi bi bi bi,ℓ′ , Ubi,ℓ′ ) associated to ′ ′ ′ ′′ , R , C , U (Gdir , P, P ′ ). By Lemma 7.3, (Gdir , Pbi , Pbi bi bi bi,ℓ′ , Ubi,ℓ′ ) is an (ℓ , 2K, m, ε , 1/2)∗ be as defined in Secbi-setup and thus also an (ℓ′ , 2K, m, ε′ , d)-bi-setup. Let BFi,dir ∗ tion 6.3. Recall from there that, for each i ≤ r3 , BFi,dir is a special factor with ∗ ) consists of all parameters (L, f ) with respect to C, P ′ in Gdir such that Fict(BFi,dir ∗ the edges in the J for all the Lf balanced exceptional systems J contained in BFi . ′ ′′ , R , C , U Thus we can apply Lemma 7.4 to (Gdir , Pbi , Pbi bi bi bi,ℓ′ , Ubi,ℓ′ ) with 2K, Lf , ε′ playing the roles of k, q, ε in order to obtain a spanning subdigraph CAdir (r) of Gdir which satisfies Lemma 7.4(i). Hence the underlying undirected graph CA(r) of


CAdir (r) satisfies Corollary 7.5(i). Indeed, to check that CA(r) and BF are edgedisjoint, by Lemma 7.4(i) it suffices to check that CA(r) avoids all edges in all the balanced exceptional systems J contained in BFi (for all i ≤ r3 ). But this follows since E(Gdir ) ⊇ E(CA(r)) consists only of AB-edges by (Sch2′ ) and since no balanced exceptional system contains an AB-edge by (BES2). Now let BF1′ , . . . , BFr′⋄ be balanced exceptional factors as described in Corollary 7.5(ii). Similarly as before, for each i ≤ r ⋄ , (BFi′ )∗dir is a special factor with parameters (1, 7) with respect to C, P in Gdir such that Fict((BFi′ )∗dir ) consists of all the edges in the J ∗ over all the 7 balanced exceptional systems J contained in BFi′ . Thus we can apply Lemma 7.4 to obtain a spanning subdigraph P CAdir (r) of Gdir which satisfies Lemma 7.4(ii)(a) and (ii)(b). Hence the underlying undirected graph P CA(r) of P CAdir (r) satisfies Corollary 7.5(ii)(a). It remains to check that Corollary 7.5(ii)(b) holds too. Thus let H be as described in Corollary 7.5(ii)(b). Let Hdir be an r-regular orientation of H. (To see that such an orientation exists, apply Petersen’s theorem to obtain a decomposition of H into 2-factors and then orient each 2-factor to obtain a (directed) 1-factor.) Let BF ∗dir be ∗ over all i ≤ r3 and let (BF ′ )∗dir be the union of the (BFi′ )∗dir the union of the BFi,dir ⋄ over all i ≤ r . Then Lemma 7.4(ii)(b) implies that Hdir + CAdir (r) + P CAdir (r) + BF ∗dir + (BF ′ )∗dir has a decomposition into s′ edge-disjoint (directed) Hamilton cycles ∗ for some balanced exceptional path C1′ , . . . , Cs′ ′ such that each Ci′ contains BEP Si,dir system BEP Si from BEPS. Let Ci be the undirected graph obtained from Ci′ − ∗ BEP Si,dir +BEP Si by ignoring the directions of all the edges. Then Proposition 6.2 (applied with G′ playing the role of G) implies that C1 , . . . , Cs′ is a decomposition of H + Grob = H + CA(r) + P CA(r) + BF + BF ′ into edge-disjoint Hamilton cycles.

8. Proof of Theorem 1.6 The proof of Theorem 1.6 is similar to that of Theorem 1.5 except that we do not need to apply the robust decomposition lemma in the proof of Theorem 1.6. For both results, we will need an approximate decomposition result (Lemma 8.1), which is stated below and proved as Lemma 3.2 in [4]. The lemma extends a suitable set of balanced exceptional systems into a set of edge-disjoint Hamilton cycles covering most edges of an almost complete and almost balanced bipartite graph. Lemma 8.1. Suppose that 0 < 1/n ≪ ε0 ≪ 1/K ≪ ρ ≪ 1 and 0 ≤ µ ≪ 1, where n, K ∈ N and K is even. Suppose that G is a graph on n vertices and P is a (K, m, ε0 )-partition of V (G). Furthermore, suppose that the following conditions hold: (a) d(w, Bi ) = (1 − 4µ ± 4/K)m and d(v, Ai ) = (1 − 4µ ± 4/K)m for all w ∈ A, v ∈ B and 1 ≤ i ≤ K. (b) There is a set J which consists of at most (1/4−µ−ρ)n edge-disjoint balanced exceptional systems with parameter ε0 in G.


(c) J has a partition into K 4 sets Ji1 ,i2 ,i3 ,i4 (one for all 1 ≤ i1 , i2 , i3 , i4 ≤ K) such that each Ji1 ,i2 ,i3 ,i4 consists of precisely |J |/K 4 (i1 , i2 , i3 , i4 )-BES with respect to P. (d) Each v ∈ A ∪ B is incident with an edge in J for at most 2ε0 n J ∈ J . Then G contains |J | edge-disjoint Hamilton cycles such that each of these Hamilton cycles contains some J ∈ J . To prove Theorem 1.6, we find a framework via Corollary 4.13. Then we choose suitable balanced exceptional systems using Corollary 5.11. Finally, we extend these into Hamilton cycles using Lemma 8.1. Proof of Theorem 1.6. Step 1: Choosing the constants and a framework. By making α smaller if necessary, we may assume that α ≪ 1. Define new constants such that 0 < 1/n0 ≪ εex ≪ ε0 ≪ ε′0 ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K ≪ α ≪ ε ≪ 1, where K ∈ N and K is even. Let G, F and D be as in Theorem 1.6. Apply Corollary 4.13 with εex , ε0 playing 1/3 the role of ε, ε∗ to find a set C1 of at most εex n edge-disjoint Hamilton cycles in F so that the graph G1 obtained from G by deleting all the edges in these Hamilton cycles 1/3 forms part of an (ε0 , ε′ , K, D1 )-framework (G1 , A, A0 , B, B0 ) with D1 ≥ D − 2εex n. ′ Moreover, F satisfies (WF5) with respect to ε and |C1 | = (D − D1 )/2.

(8.1)

In particular, this implies that δ(G1 ) ≥ D1 and that D1 is even (since D is even). Let F1 be the graph obtained from F by deleting all those edges lying on Hamilton cycles in C1 . Then δ(F1 ) ≥ δ(F ) − 2|C1 | ≥ (1/2 − 3ε1/3 ex )n.

(8.2) Let

|B| (1 − 20ε4 )D1 |A| = and tK := . K K 2K 4 By changing ε4 slightly, we may assume that tK ∈ N. Step 2: Choosing a (K, m, ε0 )-partition P. Apply Lemma 5.2 to (G1 , A, A0 , B, B0 ) with F1 , ε0 playing the roles of F , ε in order to obtain partitions A1 , . . . , AK and B1 , . . . , BK of A and B into sets of size m such that together with A0 and B0 the sets Ai and Bi form a (K, m, ε0 , ε1 , ε2 )-partition P for G1 . Note that by Lemma 5.2(ii) and since F satisfies (WF5), for all x ∈ A and 1 ≤ j ≤ K, we have m :=

dF1 (x, B) − ε1 n (WF5) dF1 (x) − ε′ n − |B0 | − ε1 n ≥ K K 1/3 (8.2) (1/2 − 3ε ex )n − 2ε1 n (8.3) ≥ (1 − 5ε1 )m. ≥ K Similarly, dF1 (y, Ai ) ≥ (1 − 5ε1 )m for all y ∈ B and 1 ≤ i ≤ K. dF1 (x, Bj )

≥


Step 3: Choosing balanced exceptional systems for the almost decomposition. Apply Corollary 5.11 to the (ε0 , ε′ , K, D1 )-framework (G1 , A, A0 , B, B0 ) with F1 , G1 , ε0 , ε′0 , D1 playing the roles of F , G, ε, ε0 , D. Let J ′ be the union of the sets Ji1 i2 i3 i4 guaranteed by Corollary 5.11. So J ′ consists of K 4 tK edge-disjoint balanced exceptional systems with parameter ε′0 in G1 (with respect to P). Let C2 denote the set of 10ε4 D1 Hamilton cycles guaranteed by Corollary 5.11. Let F2 be the subgraph obtained from F1 by deleting all the Hamilton cycles in C2 . Note that (8.4)

D2 := D1 − 2|C2 | = (1 − 20ε4 )D1 = 2K 4 tK = 2|J ′ |.

Step 4: Finding the remaining Hamilton cycles. Our next aim is to apply Lemma 8.1 with F2 , J ′ , ε′ playing the roles of G, J , ε0 . Clearly, condition (c) of Lemma 8.1 is satisfied. In order to see that condition (a) is satisfied, let µ := 1/K and note that for all w ∈ A we have (8.3) dF2 (w, Bi ) ≥ dF1 (w, Bi ) − 2|C2 | ≥ (1 − 5ε1 )m − 20ε4 D1 ≥ (1 − 1/K)m.

Similarly dF2 (v, Ai ) ≥ (1 − 1/K)m for all v ∈ B. To check condition (b), note that

D n (8.4) D2 ≤ ≤ (1/2 − α) ≤ (1/4 − µ − α/3)n. |J ′ | = 2 2 2 Thus condition (b) of Lemma 8.1 holds with α/3 playing the role of ρ. Since the edges in J ′ lie in G1 and (G1 , A, A0 , B, B0 ) is an (ε0 , ε′ , K, D1 )-framework, (FR5) implies that each v ∈ A∪B is incident with an edge in J for at most ε′ n+|V0 | ≤ 2ε′ n J ∈ J ′ . (Recall that in a balanced exceptional system there are no edges between A and B.) So condition (d) of Lemma 8.1 holds with ε′ playing the role of ε0 . So we can indeed apply Lemma 8.1 to obtainSa collection C3 of |J ′ | edge-disjoint Hamilton cycles in F2 which cover all edges of J ′ . Then C1 ∪ C2 ∪ C3 is a set of edge-disjoint Hamilton cycles in F of size D1 − D2 D2 D (8.1),(8.4) D − D1 + + = , |C1 | + |C2 | + |C3 | = 2 2 2 2 as required. 9. Proof of Theorem 1.5 As mentioned earlier, the proof of Theorem 1.5 is similar to that of Theorem 1.6 except that we will also need to apply the robust decomposition lemma. This means Steps 2–4 and Step 8 in the proof of Theorem 1.5 do not appear in the proof of Theorem 1.6. Steps 2–4 prepare the ground for the application of the robust decomposition lemma and in Step 8 we apply it to cover the leftover from the approximate decomposition step with Hamilton cycles. Steps 5–7 contain the approximate decomposition step, using Lemma 8.1. In our proof of Theorem 1.5 it will be convenient to work with an undirected version of the schemes introduced in Section 6.4. Given a graph G and partitions P and P ′ of a vertex set V , we call (G, P, P ′ ) a (K, L, m, ε0 , ε)-scheme if the following properties hold:


(Sch1) (P, P ′ ) is a (K, L, m, ε0 )-partition of V . Moreover, V (G) = A ∪ B. (Sch2) Every edge of G joins some vertex in A to some vertex in B. (Sch3) dG (v, Ai,j ) ≥ (1 − ε)m/L and dG (w, Bi,j ) ≥ (1 − ε)m/L for all v ∈ B, w ∈ A, i ≤ K and j ≤ L. We will also use the following proposition. Proposition 9.1. Suppose that K, L, n, m/L ∈ N and 0 < 1/n ≪ ε, ε0 ≪ 1. Let (G, P, P ′ ) be a (K, L, m, ε0 , ε)-scheme with |G| = n.√Then there exists an orientation Gdir of G such that (Gdir , P, P ′ ) is a [K, L, m, ε0 , 2 ε]-scheme. Proof. Randomly orient every edge in G to obtain an oriented graph Gdir . (So given any edge xy in G with probability 1/2, xy ∈ E(Gdir ) and with probability 1/2, yx ∈ E(Gdir ).) (Sch1′ ) and (Sch2′ ) follow immediately from (Sch1)√and (Sch2). Note that Fact 2.2 and (Sch3) imply that G[Ai,j , Bi′ ,j ′ ] is [1, ε]-superregular with density at least 1 − ε, for all i, i′ ≤ K and j, j ′ ≤ L. Using this, (Sch3′ ) follows easily from the large deviation bound in Proposition 2.3. (Sch4′ ) follows from Proposition 2.3 in a similar way.

Proof of Theorem 1.5. Step 1: Choosing the constants and a framework. Define new constants such that (9.1)

0 < 1/n0 ≪ εex ≪ ε∗ ≪ ε0 ≪ ε′0 ≪ ε′ ≪ ε1 ≪ ε2 ≪ ε3 ≪ ε4 ≪ 1/K2 ≪ γ ≪ 1/K1 ≪ ε′′ ≪ 1/L ≪ 1/f ≪ γ1 ≪ 1/g ≪ ε ≪ 1,

where K1 , K2 , L, f, g ∈ N and both K2 , g are even. Note that we can choose the constants such that K2 4f K1 K1 , , ∈ N. 28f gL 4gLK1 3g(g − 1) Let G and D be as in Theorem 1.5. By applying Dirac’s theorem to remove a suitable number of edge-disjoint Hamilton cycles if necessary, we may assume that D ≤ n/2. Apply Corollary 4.13 with G, εex , ε∗ , ε0 , K2 playing the roles of F , ε, ε∗ , 1/3 ε′ , K to find a set C1 of at most εex n edge-disjoint Hamilton cycles in G so that the graph G1 obtained from G by deleting all the edges in these Hamilton cycles forms part of an (ε∗ , ε0 , K2 , D1 )-framework (G1 , A, A0 , B, B0 ), where (9.2) |A|+ε0 n ≥ n/2 ≥ D1 = D−2|C1 | ≥ D−2ε1/3 ex n ≥ D−ε0 n ≥ n/2−2ε0 n ≥ |A|−2ε0 n. Note that G1 is D1 -regular and that D1 is even since D was even. Moreover, since K2 /LK1 ∈ N, (G1 , A, A0 , B, B0 ) is also an (ε∗ , ε0 , K1 L, D1 )-framework and thus an (ε∗ , ε′ , K1 L, D1 )-framework.


Let |B| |A| = , r := γm1 , r1 := γ1 m1 , r2 := 192g 3 K1 r, K1 K1 2rK1 r3 := , r ⋄ := r1 + r2 + r − (Lf − 1)r3 , L (1 − 20ε4 )D1 D4 := D1 − 2(Lf r3 + 7r ⋄ ), tK1 L := . 2(K1 L)4

m1 :=

Note that (FR4) implies m1 /L ∈ N. Moreover, (9.3)

r2 , r3 ≤ γ 1/2 m1 ≤ γ 1/3 r1 ,

r1 /2 ≤ r ⋄ ≤ 2r1 .

Further, by changing γ, γ1 , ε4 slightly, we may assume that r/K22 , r1 , tK1 L ∈ N. Since K1 /L ∈ N this implies that r3 ∈ N. Finally, note that (9.4)

(9.3) (9.2) (1 + 3ε∗ )|A| ≥ D ≥ D4 ≥ D1 − γ1 n ≥ |A| − 2γ1 n ≥ (1 − 5γ1 )|A|.

Step 2: Choosing a (K1 , L, m1 , ε0 )-partition (P1 , P1′ ). We now prepare the ground for the construction of the robustly decomposable graph Grob , which we will obtain via the robust decomposition lemma (Corollary 7.5) in Step 4. Recall that (G1 , A, A0 , B, B0 ) is an (ε∗ , ε′ , K1 L, D1 )-framework. Apply Lemma 5.2 with G1 , D1 , K1 L, ε∗ playing the roles of G, D, K, ε to obtain partitions A′1 , . . . , A′K1 L ′ of B into sets of size m1 /L such that together with A0 of A and B1′ , . . . , BK 1L ′ and B0 all these sets Ai and Bi′ form a (K1 L, m1 /L, ε∗ , ε1 , ε2 )-partition P1′ for G1 . Note that (1 − ε0 )n ≤ n − |A0 ∪ B0 | = 2K1 m1 ≤ n by (FR4). For all i ≤ K1 and all h ≤ L, let Ai,h := A′(i−1)L+h . (So this is just a relabeling of the sets S S A′i .) Define Bi,h similarly and let Ai := h≤L Ai,h and Bi := h≤L Bi,h . Let P1 := {A0 , B0 , A1 , . . . , AK1 , B1 , . . . , BK1 } denote the corresponding (K1 , m1 , ε0 )partition of V (G). Thus (P1 , P1′ ) is a (K, L, m1 , ε0 )-partition of V (G), as defined in Section 6.2. Let G2 := G1 [A, B]. We claim that (G2 , P1 , P ′ 1 ) is a (K1 , L, m1 , ε0 , ε′ )-scheme. Indeed, clearly (Sch1) and (Sch2) hold. To verify (Sch3), recall that that (G1 , A, A0 , B, B0 ) is an (ε∗ , ε0 , K1 L, D1 )-framework and so by (FR5) for all x ∈ B we have (9.2) dG2 (x, A) ≥ dG1 (x) − dG1 (x, B ′ ) − |A0 | ≥ D1 − ε0 n − |A0 | ≥ |A| − 4ε0 n

and similarly dG2 (y, B) ≥ |B| − 4ε0 n for all y ∈ A. Since ε0 ≪ ε′ /K1 L, this implies (Sch3). Step 3: Balanced exceptional systems for the robustly decomposable graph. In order to apply Corollary 7.5, we first need to construct suitable balanced exceptional systems. Apply Corollary 5.11 to the (ε∗ , ε′ , K1 L, D1 )-framework (G1 , A, A0 , B, B0 ) with G1 , K1 L, P1′ , ε∗ playing the roles of F , K, P, ε in order to obtain a set J of (K1 L)4 tK1 L edge-disjoint balanced exceptional systems in G1 with parameter ε0 such that for all 1 ≤ i′1 , i′2 , i′3 , i′4 ≤ K1 L the set J contains precisely tK1 L (i′1 , i′2 , i′3 , i′4 )-BES with respect to the partition P1′ . (Note that F in Corollary 5.11 satisfies (WF5) since G1 satisfies (FR5).) So J is the union of all the sets Ji′1 i′2 i′3 i′4 returned by Corollary 5.11. (Note that we will not use all the balanced exceptional


systems in J and we do not need to consider the Hamilton cycles guaranteed by this result. So we do not need the full strength of Corollary 5.11 at this point.) Our next aim is to choose two disjoint subsets JCA and JPCA of J with the following properties: (a) In total JCA contains Lf r3 balanced exceptional systems. For each i ≤ f and each h ≤ L, JCA contains precisely r3 (i1 , i2 , i3 , i4 )-BES of style h (with respect to the (K, L, m1 , ε0 )-partition (P1 , P1′ )) such that i1 , i2 , i3 , i4 ∈ {(i − 1)K1 /f + 2, . . . , iK1 /f }. (b) In total JPCA contains 7r ⋄ balanced exceptional systems. For each i ≤ 7, JPCA contains precisely r ⋄ (i1 , i2 , i3 , i4 )-BES (with respect to the partition P1 ) with i1 , i2 , i3 , i4 ∈ {(i − 1)K1 /7 + 2, . . . , iK1 /7}.

(Recall that we defined in Section 6.4 when an (i1 , i2 , i3 , i4 )-BES has style h with respect to a (K, L, m1 , ε0 )-partition (P1 , P1′ ).) To see that it is possible to choose JCA and JPCA , split J into two sets J1 and J2 such that both J1 and J2 contain at least tK1 L /3 (i′1 , i′2 , i′3 , i′4 )-BES with respect to P1′ , for all 1 ≤ i′1 , i′2 , i′3 , i′4 ≤ K1 L. Note that there are (K1 /f − 1)4 choices of 4-tuples (i1 , i2 , i3 , i4 ) with i1 , i2 , i3 , i4 ∈ {(i − 1)K1 /f + 2, . . . , iK1 /f }. Moreover, for each such 4-tuple (i1 , i2 , i3 , i4 ) and each h ≤ L there is one 4-tuple (i′1 , i′2 , i′3 , i′4 ) with 1 ≤ i′1 , i′2 , i′3 , i′4 ≤ K1 L and such that any (i′1 , i′2 , i′3 , i′4 )-BES with respect to P1′ is an (i1 , i2 , i3 , i4 )-BES of style h with respect to (P1 , P1′ ). Together with the fact that (9.3) (K1 /f − 1)4 tK1 L D1 1/2 ≥ ≥ γ n ≥ r3 , 3 7(Lf )4

this implies that we can choose a set JCA ⊆ J1 satisfying (a). Similarly, there are (K1 /7 − 1)4 choices of 4-tuples (i1 , i2 , i3 , i4 ) with i1 , i2 , i3 , i4 ∈ {(i − 1)K1 /7 + 2, . . . , iK1 /7}. Moreover, for each such 4-tuple (i1 , i2 , i3 , i4 ) there are L4 distinct 4-tuples (i′1 , i′2 , i′3 , i′4 ) with 1 ≤ i′1 , i′2 , i′3 , i′4 ≤ K1 L and such that any (i′1 , i′2 , i′3 , i′4 )-BES with respect to P1′ is an (i1 , i2 , i3 , i4 )-BES with respect to P1 . Together with the fact that D1 n (9.3) ⋄ (K1 /7 − 1)4 L4 tK1 L ≥ 5 ≥ ≥ r , 3 7 3 · 75

this implies that we can choose a set JPCA ⊆ J2 satisfying (b).

Step 4: Finding the robustly decomposable graph. Recall that (G2 , P1 , P1′ ) is a (K1 , L, m1 , ε0 , ε′ )-scheme. Apply Proposition 9.1 with G2 , P1 , P1′ , K1 , m1 , ε′ playing the roles of G, P, P ′ , K, m,√ε to obtain an orientation G2,dir of G2 such that (G2,dir , P1 , P1′ ) is a [K1 , L, m1 , ε0 , 2 ε′ ]-scheme. Let C = A1 B1 A2 . . . AK1 BK1 be a spanning cycle on the clusters in P1 . Our next aim is to use Lemma 6.4 in order to extend the balanced exceptional systems in JCA into r3 edge-disjoint balanced exceptional factors with parameters (L, f ) for G2,dir (with respect to C, P1′ ). For this, note that the condition on JCA in Lemma 6.4 with r3 playing the role of q is satisfied by (a). Moreover, Lr3 /m1 = 2rK1 /m1 = 2γK1 ≪ 1. Thus we can indeed apply Lemma 6.4 to (G2,dir , P1 , P1′ )


√ with JCA , 2 ε′ , K1 , r3 playing the roles of J , ε, K, q in order to obtain r3 edgedisjoint balanced exceptional factors BF1 , . . . , BFr3 with parameters (L, f ) for G2,dir ′ (with respect S to C, P1 ) such that together these balanced exceptional factors cover all edges in JCA . Let BF CA := BF1 + · · · + BFr3 . Note that m1 /4g, m1 /L ∈ N since m1 = |A|/K1 and |A| is divisible by K2 and thus m1 is divisible by 4gL (since K2 /4gLK1 ∈ N by our assumption). Furthermore, 4rK12 = 4γm1 K12 ≤ γ 1/2 m1 ≤ m1 . Thus we can apply Corollary 7.5 to the [K1 , L, m1 , ε0 , ε′′ ]-scheme (G2,dir , P1 , P1′ ) with K1 , ε′′ , g playing the roles of K, ε, ℓ′ to obtain a spanning subgraph CA(r) of G2 as described there. (Note that G2 equals the graph G′ defined in Corollary 7.5.) In particular, CA(r) is 2(r1 + r2 )-regular and edge-disjoint from BF CA . Let G3 be the graph obtained from G2 by deleting all the edges of CA(r) + BF CA . Thus G3 is obtained from G2 by deleting at most 2(r1 + r2 + r3 ) ≤ 6r1 = 6γ1 m1 edges at every vertex in A ∪ B = V (G3 ). Let G3,dir be the orientation of G3 in which every edge is oriented in the same way as in G2,dir . Then Proposition 2.1 implies that (G3,dir , P1 , P1 ) is still a [K1 , 1, m1 , ε0 , ε]-scheme. Moreover, r ⋄ (9.3) 2r1 = 2γ1 ≪ 1. ≤ m1 m1 Together with (b) this ensures that we can apply Lemma 6.4 to (G3,dir , P1 ) with P1 , JP CA , K1 , 1, 7, r ⋄ playing the roles of P, J , K, L, f , q in order to obtain r ⋄ edge-disjoint balanced exceptional factors BF1′ , . . . , BFr′⋄ with parameters (1, 7) for G3,dir (with respect these balanced exceptional factors S to C, P1 ) such that together ′ cover all edges in JPCA . Let BF PCA := BF1 + · · · + BFr′⋄ . Apply Corollary 7.5 to obtain a spanning subgraph P CA(r) of G2 as described there. In particular, P CA(r) is 10r ⋄ -regular and edge-disjoint from CA(r)+BF CA + BF PCA . Let Grob := CA(r)+P CA(r)+BF CA +BF PCA . Note that by (6.1) all the vertices in V0 := A0 ∪ B0 have the same degree r0rob := 2(Lf r3 + 7r ⋄ ) in Grob . So (9.5)

(9.3) (9.3) 7r1 ≤ r0rob ≤ 30r1 .

Moreover, (6.1) also implies that all the vertices in A ∪ B have the same degree r rob in Grob , where r rob = 2(r1 + r2 + r3 + 6r ⋄ ). So r0rob − r rob = 2 (Lf r3 + r ⋄ − (r1 + r2 + r3 )) = 2(Lf r3 + r − (Lf − 1)r3 − r3 ) = 2r. Step 5: Choosing a (K2 , m2 , ε0 )-partition P2 . We now prepare the ground for the approximate decomposition step (i.e. to apply Lemma 8.1). For this, we need to work with a finer partition of A ∪ B than the previous one (this will ensure that the leftover from the approximate decomposition step is sufficiently sparse compared to Grob ). Let G4 := G1 − Grob (where G1 was defined in Step 1) and note that (9.6)

D4 = D1 − r0rob = D1 − r rob − 2r.


So (9.7) dG4 (x) = D4 + 2r for all x ∈ A ∪ B

and

dG4 (x) = D4 for all x ∈ V0 .

(Note that D4 is even since D1 and r0rob are even.) So G4 is D4 -balanced with respect to (A, A0 , B, B0 ) by Proposition 4.1. Together with the fact that (G1 , A, A0 , B, B0 ) is an (ε∗ , ε0 , K2 , D1 )-framework, this implies that (G4 , G4 , A, A0 , B, B0 ) satisfies conditions (WF1)–(WF5) in the definition of an (ε∗ , ε0 , K2 , D4 )-weak framework. However, some vertices in A0 ∪ B0 might violate condition (WF6). (But every vertex in A ∪ B will still satisfy (WF6) with room to spare.) So we need to modify the partition of V0 = A0 ∪ B0 to obtain a new weak framework. Consider a partition A∗0 , B0∗ of A0 ∪ B0 which maximizes the number of edges in G4 between A∗0 ∪ A and B0∗ ∪ B. Then dG4 (v, A∗0 ∪ A) ≤ dG4 (v)/2 for all v ∈ A∗0 since otherwise A∗0 \ {v}, B0∗ ∪ {v} would be a better partition of A0 ∪ B0 . Similarly dG4 (v, B0∗ ∪ B) ≤ dG4 (v)/2 for all v ∈ B0∗ . Thus (WF6) holds in G4 (with respect to the partition A ∪ A∗0 and B ∪ B0∗ ). Moreover, Proposition 4.2 implies that G4 is still D4 -balanced with respect to (A, A∗0 , B, B0∗ ). Furthermore, with (FR3) and (FR4) applied to G1 , we obtain eG4 (A ∪ A∗0 ) ≤ eG1 (A ∪ A0 ) + |A∗0 ||A ∪ A∗0 | ≤ ε0 n2 and similarly eG4 (B ∪ B0∗ ) ≤ ε0 n2 . Finally, every vertex in A ∪ B has internal degree at most ε0 n + |A0 ∪ B0 | ≤ 2ε0 n in G4 (with respect to the partition A ∪ A∗0 and B ∪ B0∗ ). Altogether this implies that (G4 , G4 , A, A∗0 , B, B0∗ ) is an (ε0 , 2ε0 , K2 , D4 )weak framework and thus also an (ε0 , ε′ , K2 , D4 )-weak framework. Without loss of generality we may assume that |A∗0 | ≥ |B0∗ |. Apply Lemma 4.12 to the (ε0 , ε′ , K2 , D4 )-weak framework (G4 , G4 , A, A∗0 , B, B0∗ ) to find a set C2 of |C2 | ≤ ε0 n edge-disjoint Hamilton cycles in G4 so that the graph G5 obtained from G4 by deleting all the edges of these Hamilton cycles forms part of an (ε0 , ε′ , K2 , D5 )framework (G5 , A, A∗0 , B, B0∗ ), where (9.8)

D5 = D4 − 2|C2 | ≥ D4 − 2ε0 n.

Since D4 is even, D5 is even. Further, (9.9) (9.7) dG5 (x) = D5 +2r for all x ∈ A∪B

and

(9.7) dG5 (x) = D5 for all x ∈ A∗0 ∪B0∗ .

Choose an additional constant ε′4 such that ε3 ≪ ε′4 ≪ 1/K2 and so that tK2 :=

(1 − 20ε′4 )D5 ∈ N. 2K24

Now apply Lemma 5.2 to (G5 , A, A∗0 , B, B0∗ ) with D5 , K2 , ε0 playing the roles of D, K, ε in order to obtain partitions A1 , . . . , AK2 and B1 , . . . , BK2 of A and B into sets of size (9.10)

m2 := |A|/K2

such that together with A∗0 and B0∗ the sets Ai and Bi form a (K2 , m2 , ε0 , ε1 , ε2 )partition P2 for G5 . (Note that the previous partition of A and B plays no role in the subsequent argument, so denoting the clusters in P2 by Ai and Bi again will cause no notational conflicts.)


Step 6: Balanced exceptional systems for the approximate decomposition. In order to apply Lemma 8.1, we first need to construct suitable balanced exceptional systems. Apply Corollary 5.11 to the (ε0 , ε′ , K2 , D5 )-framework (G5 , A, A∗0 , B, B0∗ ) with G5 , ε0 , ε′0 , ε′4 , K2 , D5 , P2 playing the roles of F , ε, ε0 , ε4 , K, D, P. (Note that since we are letting G5 play the role of F , condition (WF5) in the corollary immediately follows from (FR5).) Let J ′ be the union of the sets Ji1 i2 i3 i4 guaranteed by Corollary 5.11. So J ′ consists of K24 tK2 edge-disjoint balanced exceptional systems with parameter ε′0 in G5 (with respect to P2 ). Let C3 denote the set of Hamilton cycles guaranteed by Corollary 5.11. So |C3 | = 10ε′4 D5 . Let G6 be the subgraph obtained from G5 by deleting all those edges lying in the Hamilton cycles from C3 . Set D6 := D5 − 2|C3 |. So (9.11) (9.9) (9.9) dG6 (x) = D6 + 2r for all x ∈ A ∪ B and dG6 (x) = D6 for all x ∈ V0 . (Note that V0 = A0 ∪ B0 = A∗0 ∪ B0∗ .) Let G′6 denote the subgraph of G6 obtained by deleting all those edges lying in the balanced exceptional systems from J ′ . Thus G′6 = G⋄ , where G⋄ is as defined in Corollary 5.11(iv). In particular, V0 is an isolated set in G′6 and G′6 is bipartite with vertex classes A ∪ A∗0 and B ∪ B0∗ (and thus also bipartite with vertex classes A′ = A ∪ A0 and B ′ = B ∪ B0 ). Consider any vertex v ∈ V0 . Then v has degree D5 in G5 , degree two in each Hamilton cycle from C3 , degree two in each balanced exceptional system from J ′ and degree zero in G′6 . Thus (9.9) D6 + 2|C3 | = D5 = dG5 (v) = 2|C3 | + 2|J ′ | + dG′6 (v) = 2|C3 | + 2|J ′ |

and so D6 = 2|J ′ |.

(9.12)

Step 7: Approximate Hamilton cycle decomposition. Our next aim is to apply Lemma 8.1 with G6 , P2 , K2 , m2 , J ′ , ε′ playing the roles of G, P, K, m, J , ε0 . Clearly, condition (c) of Lemma 8.1 is satisfied. In order to see that condition (a) is satisfied, let µ := (r0rob − 2r)/4K2 m2 and note that 0≤

γ1 m 1 7r1 − 2r (9.5) (9.5) 30r1 30γ1 ≤ ≤ ≪ 1. ≤ µ ≤ 4K2 m2 4K2 m2 4K2 m2 K1

Recall that every vertex v ∈ B satisfies (9.9) (9.6),(9.8) (9.2) dG5 (v) = D5 + 2r = D1 − r0rob + 2r ± 2ε0 n = |A| − r0rob + 2r ± 4ε0 n.

Moreover, dG5 (v, A) = dG5 (v) − dG5 (v, B ∪ B0∗ ) − |A∗0 | ≥ dG5 (v) − 2ε′ n,


where the last inequality holds since (G5 , A, A∗0 , B, B0∗ ) is an (ε0 , ε′ , K2 , D5 )-framework (c.f. conditions (FR4) and (FR5)). Together with the fact that P2 is a (K2 , m2 , ε0 , ε1 , ε2 )partition for G5 (c.f. condition (P2)), this implies that |A| − r0rob + 2r ± 2ε1 n r0rob − 2r dG5 (v, A) ± ε1 n = = 1− ± 5ε1 m2 dG5 (v, Ai ) = K2 K2 K2 m2 = (1 − 4µ ± 5ε1 )m2 = (1 − 4µ ± 1/K2 )m2 . Recall that G6 is obtained from G5 by deleting all those edges lying in the Hamilton cycles in C3 and that (9.4) (9.10 ) |C3 | = 10ε′4 D5 ≤ 10ε′4 D4 ≤ 11ε′4 |A| ≤ m2 /K2 .

Altogether this implies that dG6 (v, Ai ) = (1 − 4µ ± 4/K2 )m2 . Similarly one can show that dG6 (w, Bj ) = (1 − 4µ ± 4/K2 )m2 for all w ∈ A. So condition (a) of Lemma 8.1 holds. To check condition (b), note that 1 D4 (9.6) D1 − r0rob n γ ′ (9.12) D6 |J | = ≤ ≤ − µ · 2K2 m2 − r ≤ −µ− n. ≤ 2 2 2 4 4 3K1 Thus condition (b) of Lemma 8.1 holds with γ/3K1 playing the role of ρ. Since the edges in J ′ lie in G5 and (G5 , A, A∗0 , B, B0∗ ) is an (ε0 , ε′ , K2 , D5 )-framework, (FR5) implies that each v ∈ A ∪ B is incident with an edge in J for at most ε′ n + |V0 | ≤ 2ε′ n of the J ∈ J ′ . (Recall that in a balanced exceptional system there are no edges between A and B.) So condition (d) of Lemma 8.1 holds with ε′ playing the role of ε0 . So we can indeed apply Lemma 8.1 to obtainSa collection C4 of |J ′ | edge-disjoint Hamilton cycles in G6 which cover all edges of J ′ .

Step 8: Decomposing the leftover and the robustly decomposable graph. Finally, we can apply the ‘robust decomposition property’ of Grob guaranteed by Corollary 7.5 to obtain a Hamilton decomposition of the leftover from the previous step together with Grob . To achieve this, let H ′ denote the subgraph of G6 obtained by deleting all those edges lying in the Hamilton cycles from C4 . Thus (9.11) and (9.12) imply that every vertex in V0 is isolated in H ′ while every vertex v ∈ A ∪ B has degree dG6 (v) − 2|J ′ | = D6 + 2r − 2|J ′ | = 2r in H ′ (the last equality follows from (9.12)). Moreover, H ′ [A] S and H ′ [B] contain no edges. (This holds since H ′ is a spanning subgraph of G6 − J ′ = G′6 and since we have already seen that G′6 is bipartite with vertex classes A′ and B ′ .) Now let H := H ′ [A, B]. Then Corollary 7.5(ii)(b) implies that H + Grob has a Hamilton decomposition. Let C5 denote the set of Hamilton cycles thus obtained. Note that H + Grob is a spanning subgraph of G which contains all edges of G which were not covered by C1 ∪ C2 ∪ C3 ∪ C4 . So C1 ∪ C2 ∪ C3 ∪ C4 ∪ C5 is a Hamilton decomposition of G.


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[27] M.J. Plantholt and S.K. Tipnis, All regular multigraphs of even order and high degree are 1-factorable, Electron. J. Combin. 8 (2001), R41. [28] M. Stiebitz, D. Scheide, B. Toft and L.M. Favrholdt, Graph Edge Coloring: Vizing’s Theorem and Goldberg’s Conjecture, Wiley 2012. [29] E. Vaughan, An asymptotic version of the multigraph 1-factorization conjecture, J. Graph Theory 72 (2013), 19–29. [30] D.B. West, Introduction to Graph Theory (2nd Edition), Pearson 2000. Béla Csaba Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértan´ uk tere 1. Hungary E-mail address: [email protected] Daniela K¨ uhn, Allan Lo, Deryk Osthus School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK E-mail addresses: {d.kuhn, s.a.lo, d.osthus}@bham.ac.uk Andrew Treglown School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS UK E-mail addresses: [email protected]

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